Solving simple linear equations. What is an equation? How to solve equations
What is an equation?
An equation is one of the cornerstone concepts of all mathematics. Both school and higher education. It makes sense to figure it out, right? Moreover, this is a very simple concept. See for yourself below. :) So what is the equation?
The fact that this word has the same root as the words “equal”, “equality”, I think, does not raise any objections from anyone. An equation is two mathematical expressions connected by an equal sign “=”. But... not just any. And those in which (at least one) contains unknown quantity . Or in another way variable quantity . Or simply “variable” for short. There can be one or more variables. In school mathematics, equations with one variable. Which is usually denoted by the letterx . Or other last letters of the Latin alphabet -y , z , t and so on.
For now we will also consider equations with one variable. With two or more variables - in a special lesson.
What does it mean to solve an equation?
Go ahead. The variable in the expressions included in the equation can take any valid value. That's why it's variable. :) For some values of the variable the correct equality is obtained, but for others it is not. Solve the equation- this means finding all such values of the variable, when substituting them into original the equation turns out true equality . Or, more scientifically, identity. For example, 5=5, 0=0, -10=-10. And so on. :) Or prove that such variable values do not exist.
I specifically focus on the word “original”. Why will become clear below.
These very values of the variable, upon substitution of which the equation turns into an identity, are called very beautifully - roots of the equation. If it is proven that there are no such values, then in this case they say that the equation has no roots.
Why are equations needed?
Why do we need equations? First of all, equations are a very powerful and most versatile tool for problem solving . Very different. :) At school, as a rule, they work with word problems. These are tasks on movement, on work, on percentages and many, many others. However, the use of equations is not limited to school problems about swimming pools, pipes, trains and stools. :)
Without the ability to compose and solve equations, it is impossible to solve any serious scientific problem - physical, engineering or economic. For example, calculate where a rocket will hit. Or answer the question whether some important structure (an elevator or a bridge, for example) will or will not withstand the load. Or predict the weather, rise (or fall) in prices or income...
In general, the equation is a key figure in solving a wide variety of computational problems.
What are the equations?
There are countless equations in mathematics. Various types. However, all equations can be divided into only 4 classes:
1) Linear,
2) Square,
3) Fractional (or fractional-rational),
4) Others.
Different types of equations require different approaches to their solution: linear equations are solved in one way, quadratic equations in another, fractional equations in a third, trigonometric, logarithmic, exponential and others are also solved using their own methods.
There are, of course, more other equations. These are irrational, trigonometric, exponential, logarithmic, and many other equations. And even differential equations (for students), where the unknown is not a number, but function. Or even a whole family of functions. :) In the corresponding lessons we will analyze in detail all these types of equations. And here we have basic techniques that are applicable to solve absolutely any(yes, any!) equations. These techniques are called equivalent transformations of equations . There are only two of them. And there is no way around them. So let's get acquainted!
How to solve equations? Identical (equivalent) transformations of equations.
Solution any equation consists in a step-by-step transformation of the expressions included in it. But not just any transformations, but such that the essence of the whole equation has not changed. Despite the fact that after each transformation the equation will change and ultimately become completely different from the original one. Such transformations in mathematics are called equivalent or identical . Among the whole variety of identical transformations of equations, one stands out two basic. We will talk about them. Yes, yes, only two! And each of them deserves special attention. Applying these two identical transformations in one order or another guarantees success in solving 99% of all equations.
So, let's get acquainted!
First identity transformation:
You can add (or subtract) any (but identical!) number or expression (including those with a variable) to both sides of the equation.
The essence of the equation will remain the same. You apply this transformation everywhere, naively thinking that you are transferring some terms from one part of the equation to another, changing the sign. :)
For example, this cool equation:
There’s nothing to think about here: move the minus three to the right, changing the minus to a plus:
But what is really happening? But in reality you add three to both sides of the equation! Like this:
The essence of the entire equation does not change when adding three to both sides. On the left there remains a pure X (which is what we, in fact, are trying to achieve), and on the right - whatever happens.
Transferring terms from one part to another is abridged version first identity transformation. The only mistake you can make here is to forget to change the sign when transferring. For example, this equation:
It's not a complicated matter. We work directly according to the spell: with X's to the left, without X's to the right. What term with X is on the right? What? 2x? Wrong! On the right we have -2x (minus two x's)! Therefore, this term will be transferred to the left side with a plus :
Half the battle is done, the X's have been collected on the left. All that remains is to move the unit to the right. Again the question is - with what sign? There is nothing written on the left before the unit, which means that it is meant to be preceded by plus. Therefore, the 1 will move to the right with a minus:
That's almost all. On the left we present similar ones, and on the right we count them. And we get:
Now let’s analyze our machinations with transferring terms. What did we do when we moved -2x to the left? Yes! We added to both parts of our evil equation the expression is 2x. I told you that we have the right to add (subtract) any number and even an expression with an X! As long as it's the same thing. :) And when did you move the 1 to the right? Absolutely right! We subtracted from both sides of the equation one. That's all.) That's the whole point of the first equivalent transformation.
Or this example for high school students:
The equation is logarithmic. So what? Who cares? Anyway, the first step is to do a basic identity transformation - we move the term with the variable (that is, -log 3 x) to the left, and we move the numerical expression log 3 4 to the right. With a change of sign, of course:
That's all. Anyone who is familiar with logarithms will complete the equation in his head and get:
What? Do you want sines? Please, here are the sines:
We perform the first identical transformation again - we transfer sin x to the left (with a minus), and move -1/4 to the right (with a plus):
We have obtained the simplest trigonometric equation with sine, which is also not difficult for those in the know to solve.
See how universal the first equivalent transformation is! It is found everywhere and everywhere and there is no way to get around it. Therefore, you need to be able to do it automatically. The main thing is not to forget to change the sign when transferring! We continue to get acquainted with identical transformations of equations.)
Second identity transformation:
Both sides of the equation can be multiplied (divided) by the same non-zero number or expression.
We also constantly use this identical transformation when some coefficients in the equation interfere with us and we want to get rid of them. Safe for the equation itself. :) For example, this evil equation:
It is clear to everyone here that x = 3. How did you guess? Did you pick it up? Or did you point your finger at the sky and guess?
In order not to select and guess (we are after all mathematicians, not fortune tellers :)), you need to understand that you are simply divided both sides of the equation for a four. Which is what bothers us.
Like this:
This division stick means that they are divided by four. both parts our equation. The entire left side and the entire right side:
On the left, the fours are safely reduced and the x remains in splendid isolation. And on the right, when dividing 12 by 4, the result is, naturally, three. :)
Or this equation:
What to do with one seventh? Move right? Nope, you can't! One seventh is associated with x multiplication. The coefficient, you understand. :) You cannot separate the coefficient and move it separately from the X. Only the entire expression (1/7)x. But there’s no need. :) Let's remember about multiplication/division again. What's stopping us? The fraction is 1/7, isn't it? So let's get rid of it. How? And as a result of what action do we lose the fraction? Our fraction disappears when multiplication by a number equal to its denominator! So let's multiply both sides of our equation by 7:
On the left, the sevens will be reduced and just a lone X will remain, and on the right, if you remember the multiplication table, you get 21:
Now an example for high school students:
To get to x and thereby solve our evil trigonometric equation, we must first obtain a pure cosine on the left, without any coefficients. But the deuce gets in the way. :) So we divide the entire left side by 2:
But then the right side will also have to be divided by two: this is already required by MATHEMATICS. Divide:
We received the table value of the cosine on the right. And now the equation is solved for the sweet soul.)
Is everything clear with multiplication/division? Great! But… attention! This transformation, despite its simplicity, contains a source of very annoying errors! It's called loss of roots And acquisition of foreign roots .
I already said above that both sides of the equation can be multiplied (divided) by any number or expression with x. But with one important caveat: the expression by which we multiply (divide) must be different from zero . It is this point, which many simply ignore at first, that leads to such unfortunate mistakes. Actually, the meaning of this restriction is clear: multiplying by zero is stupid, and dividing is generally not allowed. Let's figure out what's what? Let's start with division and root loss .
Let's say we have this equation:
Here you are really itching to take and divide both sides of the equation by a common bracket (x-1):
Let's say the Unified State Exam task says to find the sum of the roots of this equation. What will we write in response? Three? If you decide that it's a three, then you were ambushed. Called “root loss.” :) What's the matter?
Let's open the brackets in the original equation and collect everything on the left:
We got the classic quadratic equation. We solve through the discriminant (or through Vieta’s theorem) and get two roots:
Therefore, the sum of the roots is 1+3 = 4. Four, not three! Where did our root “disappear”?
x = 1
With the first solution? And our one disappeared just when we were dividing both parts by brackets (x-1). Why did it happen? And all because at x = 1 this very bracket (x-1) is reset to zero. And we have the right to divide only by non-zero expression! How could the loss of this root be avoided? And loss of roots in general? To do this, firstly, before dividing by some expression with an x, we always add the condition that this expression is different from zero. And we find zeros of this expression. Like this (using our equation as an example):
And secondly, so that some roots do not disappear during the division process, we must separately check as candidates for roots All zeros of our expression (the one we are dividing by). How? Just put them in original equation and count. In our case, we check one:
Everything is fair. So, one is the root!
In general, in the future, always try to avoid divisions to the expression with an X. Losing roots is a very dangerous and annoying thing! Use any other methods - opening brackets and especially factorization. Factoring is the easiest and safest way to avoid losing roots. To do this, we collect everything on the left, then we take the common factor (which we want to “reduce” by) out of brackets, factor it into factors and then equate each resulting factor to zero. For example, our equation could be solved quite harmlessly not only by reduction to a quadratic, but also by factorization. See for yourself:
Move the entire expression (x-1) to the left. With a minus sign:
We take (x-1) out of brackets as a common factor and factorize it:
The product is zero when at least one of the factors is zero. Now we equate (in our minds!) each bracket to zero and get our legal two roots:
And not a single root was lost!
Let's now look at the opposite situation - acquisition of foreign roots. This situation occurs when multiplication both sides of the equation to the expression with x. It often occurs when solving fractional rational equations. For example, this simple equation:
It’s a familiar matter - we multiply both sides by the denominator to get rid of the fraction and get a ruler equation:
We equate each factor to zero and get two roots:
Everything seems to be fine. But let's try to do a basic check. And if at x = 0 everything will grow together nicely, we get the identity 2=2, then when x = 1 This will result in division by zero. What you absolutely cannot do. One is not suitable as the root of our equation. In such cases it is said that x = 1- so-called extraneous root . One is the root of our new equation without a fraction x(x-1) = 0, But is not root original fractional equation. How does this foreign root appear? It appears when both sides are multiplied by the denominator x-1. which at x = 1 just goes to zero! And we have the right to multiply only by an expression other than zero!
How to be? Don't multiply at all? Then we won’t be able to solve anything at all. Should I check every time? Can. But it is often labor-intensive if the initial equation is too convoluted. In such cases, three magic letters come to the rescue - ODZ. ABOUT area D omitted Z accomplishments. And in order to exclude the appearance of extraneous roots, when multiplying by an expression with an X, you must always additionally write down the ODZ. In our case:
Now, with this limitation, you can safely multiply both sides by the denominator. We will exclude all harmful consequences from such multiplication (i.e. extraneous roots) according to the DZ. And we will mercilessly throw away our one.
So, the appearance of extraneous roots is not as dangerous as the loss: ODZ is a powerful thing. And tough. She will always weed out everything unnecessary. :) ODZ and I will be friends and will get to know each other in more detail in a separate lesson.
That's all the identical transformations.) Only two. However, an inexperienced student may have some difficulties associated with sequence their applications: in some examples they start with multiplication (or division), in others - with transfer. For example, this linear equation:
Where to start? You can start with the transfer:
Or you can first divide both parts by five, and then transfer. Then the numbers will become simpler and it will be easier to count:
As we see, both ways are possible. So the question arises for some students: “Which is correct?” Answer: “In every way correct!” Whichever is more convenient for you. :) As long as your actions do not contradict the rules of mathematics. And the sequence of these very actions depends solely on the personal preferences and habits of the decider. However, with experience, such questions will disappear by themselves, and in the end it will not be mathematics that will command you, but you will command mathematics. :)
In conclusion, I would like to say separately about the so-called conditionally identical transformations, valid for some conditions. For example, raising both sides of an equation to the same power. Or extracting the root from both parts. If the exponent is odd, then there are no restrictions - construct and extract without fear. But if it is even, then such a transformation will be identical only if both sides of the equation are non-negative. We will talk about these pitfalls in detail in the topic about irrational equations.
Quadratic equations are studied in 8th grade, so there is nothing complicated here. The ability to solve them is absolutely necessary.
A quadratic equation is an equation of the form ax 2 + bx + c = 0, where the coefficients a, b and c are arbitrary numbers, and a ≠ 0.
Before studying specific solution methods, note that all quadratic equations can be divided into three classes:
- Have no roots;
- Have exactly one root;
- They have two different roots.
This is an important difference between quadratic equations and linear ones, where the root always exists and is unique. How to determine how many roots an equation has? There is a wonderful thing for this - discriminant.
Discriminant
Let the quadratic equation ax 2 + bx + c = 0 be given. Then the discriminant is simply the number D = b 2 − 4ac.
You need to know this formula by heart. Where it comes from is not important now. Another thing is important: by the sign of the discriminant you can determine how many roots a quadratic equation has. Namely:
- If D< 0, корней нет;
- If D = 0, there is exactly one root;
- If D > 0, there will be two roots.
Please note: the discriminant indicates the number of roots, and not at all their signs, as for some reason many people believe. Take a look at the examples and you will understand everything yourself:
Task. How many roots do quadratic equations have:
- x 2 − 8x + 12 = 0;
- 5x 2 + 3x + 7 = 0;
- x 2 − 6x + 9 = 0.
Let's write out the coefficients for the first equation and find the discriminant:
a = 1, b = −8, c = 12;
D = (−8) 2 − 4 1 12 = 64 − 48 = 16
So the discriminant is positive, so the equation has two different roots. We analyze the second equation in a similar way:
a = 5; b = 3; c = 7;
D = 3 2 − 4 5 7 = 9 − 140 = −131.
The discriminant is negative, there are no roots. The last equation left is:
a = 1; b = −6; c = 9;
D = (−6) 2 − 4 1 9 = 36 − 36 = 0.
The discriminant is zero - the root will be one.
Please note that coefficients have been written down for each equation. Yes, it’s long, yes, it’s tedious, but you won’t mix up the odds and make stupid mistakes. Choose for yourself: speed or quality.
By the way, if you get the hang of it, after a while you won’t need to write down all the coefficients. You will perform such operations in your head. Most people start doing this somewhere after 50-70 solved equations - in general, not that much.
Roots of a quadratic equation
Now let's move on to the solution itself. If the discriminant D > 0, the roots can be found using the formulas:
Basic formula for the roots of a quadratic equation
When D = 0, you can use any of these formulas - you will get the same number, which will be the answer. Finally, if D< 0, корней нет — ничего считать не надо.
- x 2 − 2x − 3 = 0;
- 15 − 2x − x 2 = 0;
- x 2 + 12x + 36 = 0.
First equation:
x 2 − 2x − 3 = 0 ⇒ a = 1; b = −2; c = −3;
D = (−2) 2 − 4 1 (−3) = 16.
D > 0 ⇒ the equation has two roots. Let's find them:
Second equation:
15 − 2x − x 2 = 0 ⇒ a = −1; b = −2; c = 15;
D = (−2) 2 − 4 · (−1) · 15 = 64.
D > 0 ⇒ the equation again has two roots. Let's find them
\[\begin(align) & ((x)_(1))=\frac(2+\sqrt(64))(2\cdot \left(-1 \right))=-5; \\ & ((x)_(2))=\frac(2-\sqrt(64))(2\cdot \left(-1 \right))=3. \\ \end(align)\]
Finally, the third equation:
x 2 + 12x + 36 = 0 ⇒ a = 1; b = 12; c = 36;
D = 12 2 − 4 1 36 = 0.
D = 0 ⇒ the equation has one root. Any formula can be used. For example, the first one:
As you can see from the examples, everything is very simple. If you know the formulas and can count, there will be no problems. Most often, errors occur when substituting negative coefficients into the formula. Here again, the technique described above will help: look at the formula literally, write down each step - and very soon you will get rid of mistakes.
Incomplete quadratic equations
It happens that a quadratic equation is slightly different from what is given in the definition. For example:
- x 2 + 9x = 0;
- x 2 − 16 = 0.
It is easy to notice that these equations are missing one of the terms. Such quadratic equations are even easier to solve than standard ones: they don’t even require calculating the discriminant. So, let's introduce a new concept:
The equation ax 2 + bx + c = 0 is called an incomplete quadratic equation if b = 0 or c = 0, i.e. the coefficient of the variable x or the free element is equal to zero.
Of course, a very difficult case is possible when both of these coefficients are equal to zero: b = c = 0. In this case, the equation takes the form ax 2 = 0. Obviously, such an equation has a single root: x = 0.
Let's consider the remaining cases. Let b = 0, then we obtain an incomplete quadratic equation of the form ax 2 + c = 0. Let us transform it a little:
Since the arithmetic square root exists only of a non-negative number, the last equality makes sense only for (−c /a) ≥ 0. Conclusion:
- If in an incomplete quadratic equation of the form ax 2 + c = 0 the inequality (−c /a) ≥ 0 is satisfied, there will be two roots. The formula is given above;
- If (−c /a)< 0, корней нет.
As you can see, a discriminant was not required—there are no complex calculations at all in incomplete quadratic equations. In fact, it is not even necessary to remember the inequality (−c /a) ≥ 0. It is enough to express the value x 2 and see what is on the other side of the equal sign. If there is a positive number, there will be two roots. If it is negative, there will be no roots at all.
Now let's look at equations of the form ax 2 + bx = 0, in which the free element is equal to zero. Everything is simple here: there will always be two roots. It is enough to factor the polynomial:
Taking the common factor out of bracketsThe product is zero when at least one of the factors is zero. This is where the roots come from. In conclusion, let’s look at a few of these equations:
Task. Solve quadratic equations:
- x 2 − 7x = 0;
- 5x 2 + 30 = 0;
- 4x 2 − 9 = 0.
x 2 − 7x = 0 ⇒ x · (x − 7) = 0 ⇒ x 1 = 0; x 2 = −(−7)/1 = 7.
5x 2 + 30 = 0 ⇒ 5x 2 = −30 ⇒ x 2 = −6. There are no roots, because a square cannot be equal to a negative number.
4x 2 − 9 = 0 ⇒ 4x 2 = 9 ⇒ x 2 = 9/4 ⇒ x 1 = 3/2 = 1.5; x 2 = −1.5.
The online equation solving service will help you solve any equation. Using our website, you will receive not just the answer to the equation, but also see a detailed solution, that is, a step-by-step display of the process of obtaining the result. Our service will be useful to high school students and their parents. Students will be able to prepare for tests and exams, test their knowledge, and parents will be able to monitor the solution of mathematical equations by their children. The ability to solve equations is a mandatory requirement for schoolchildren. The service will help you educate yourself and improve your knowledge in the field of mathematical equations. With its help, you can solve any equation: quadratic, cubic, irrational, trigonometric, etc. The benefits of the online service are priceless, because in addition to the correct answer, you receive a detailed solution to each equation. Benefits of solving equations online. You can solve any equation online on our website absolutely free. The service is completely automatic, you don’t have to install anything on your computer, you just need to enter the data and the program will give you a solution. Any errors in calculations or typos are excluded. With us, solving any equation online is very easy, so be sure to use our site to solve any kind of equations. You only need to enter the data and the calculation will be completed in a matter of seconds. The program works independently, without human intervention, and you receive an accurate and detailed answer. Solution of the equation in general form. In such an equation, the variable coefficients and the desired roots are interconnected. The highest power of a variable determines the order of such an equation. Based on this, various methods and theorems are used for equations to find solutions. Solving equations of this type means finding the required roots in general form. Our service allows you to solve even the most complex algebraic equation online. You can obtain both a general solution to the equation and a particular one for the numerical values of the coefficients you specify. To solve an algebraic equation on the website, it is enough to correctly fill out only two fields: the left and right sides of the given equation. Algebraic equations with variable coefficients have an infinite number of solutions, and by setting certain conditions, partial ones are selected from the set of solutions. Quadratic equation. The quadratic equation has the form ax^2+bx+c=0 for a>0. Solving quadratic equations involves finding the values of x at which the equality ax^2+bx+c=0 holds. To do this, find the discriminant value using the formula D=b^2-4ac. If the discriminant is less than zero, then the equation has no real roots (the roots are from the field of complex numbers), if it is equal to zero, then the equation has one real root, and if the discriminant is greater than zero, then the equation has two real roots, which are found by the formula: D = -b+-sqrt/2a. To solve a quadratic equation online, you just need to enter the coefficients of the equation (integers, fractions or decimals). If there are subtraction signs in an equation, you must put a minus sign in front of the corresponding terms of the equation. You can solve a quadratic equation online depending on the parameter, that is, the variables in the coefficients of the equation. Our online service for finding general solutions copes well with this task. Linear equations. To solve linear equations (or systems of equations), four main methods are used in practice. We will describe each method in detail. Substitution method. Solving equations using the substitution method requires expressing one variable in terms of the others. After this, the expression is substituted into other equations of the system. Hence the name of the solution method, that is, instead of a variable, its expression is substituted through the remaining variables. In practice, the method requires complex calculations, although it is easy to understand, so solving such an equation online will help save time and make calculations easier. You just need to indicate the number of unknowns in the equation and fill in the data from the linear equations, then the service will make the calculation. Gauss method. The method is based on the simplest transformations of the system in order to arrive at an equivalent triangular system. From it, the unknowns are determined one by one. In practice, you need to solve such an equation online with a detailed description, thanks to which you will have a good understanding of the Gaussian method for solving systems of linear equations. Write down the system of linear equations in the correct format and take into account the number of unknowns in order to accurately solve the system. Cramer's method. This method solves systems of equations in cases where the system has a unique solution. The main mathematical action here is the calculation of matrix determinants. Solving equations using the Cramer method is carried out online, you receive the result instantly with a complete and detailed description. It is enough just to fill the system with coefficients and select the number of unknown variables. Matrix method. This method consists of collecting the coefficients of the unknowns in matrix A, the unknowns in column X, and the free terms in column B. Thus, the system of linear equations is reduced to a matrix equation of the form AxX=B. This equation has a unique solution only if the determinant of matrix A is different from zero, otherwise the system has no solutions, or an infinite number of solutions. Solving equations using the matrix method involves finding the inverse matrix A.
Solving exponential equations. Examples.
Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)
What's happened exponential equation? This is an equation in which the unknowns (x's) and expressions with them are in indicators some degrees. And only there! It is important.
There you are examples of exponential equations:
3 x 2 x = 8 x+3
Note! In the bases of degrees (below) - only numbers. IN indicators degrees (above) - a wide variety of expressions with an X. If, suddenly, an X appears in the equation somewhere other than an indicator, for example:
this will already be an equation of mixed type. Such equations do not have clear rules for solving them. We will not consider them for now. Here we will deal with solving exponential equations in its purest form.
In fact, even pure exponential equations are not always solved clearly. But there are certain types of exponential equations that can and should be solved. These are the types we will consider.
Solving simple exponential equations.
First, let's solve something very basic. For example:
Even without any theories, by simple selection it is clear that x = 2. Nothing more, right!? No other value of X works. Now let's look at the solution to this tricky exponential equation:
What have we done? We, in fact, simply threw out the same bases (triples). Completely thrown out. And, the good news is, we hit the nail on the head!
Indeed, if in an exponential equation there are left and right the same numbers in any powers, these numbers can be removed and the exponents can be equalized. Mathematics allows. It remains to solve a much simpler equation. Great, right?)
However, let us remember firmly: You can remove bases only when the base numbers on the left and right are in splendid isolation! Without any neighbors and coefficients. Let's say in the equations:
2 x +2 x+1 = 2 3, or
twos cannot be removed!
Well, we have mastered the most important thing. How to move from evil exponential expressions to simpler equations.
"Those are the times!" - you say. “Who would give such a primitive lesson on tests and exams!?”
I have to agree. Nobody will. But now you know where to aim when solving tricky examples. It must be brought to the form where the same base number is on the left and right. Then everything will be easier. Actually, this is a classic of mathematics. We take the original example and transform it to the desired one us mind. According to the rules of mathematics, of course.
Let's look at examples that require some additional effort to reduce them to the simplest. Let's call them simple exponential equations.
Solving simple exponential equations. Examples.
When solving exponential equations, the main rules are actions with degrees. Without knowledge of these actions nothing will work.
To actions with degrees, one must add personal observation and ingenuity. Do we need the same base numbers? So we look for them in the example in explicit or encrypted form.
Let's see how this is done in practice?
Let us be given an example:
2 2x - 8 x+1 = 0
The first keen glance is at grounds. They... They are different! Two and eight. But it’s too early to become discouraged. It's time to remember that
Two and eight are relatives in degree.) It is quite possible to write:
8 x+1 = (2 3) x+1
If we recall the formula from operations with degrees:
(a n) m = a nm ,
this works out great:
8 x+1 = (2 3) x+1 = 2 3(x+1)
The original example began to look like this:
2 2x - 2 3(x+1) = 0
We transfer 2 3 (x+1) to the right (no one has canceled the elementary operations of mathematics!), we get:
2 2x = 2 3(x+1)
That's practically all. Removing the bases:
We solve this monster and get
This is the correct answer.
In this example, knowing the powers of two helped us out. We identified in eight there is an encrypted two. This technique (encoding common bases under different numbers) is a very popular technique in exponential equations! Yes, and in logarithms too. You must be able to recognize powers of other numbers in numbers. This is extremely important for solving exponential equations.
The fact is that raising any number to any power is not a problem. Multiply, even on paper, and that’s it. For example, anyone can raise 3 to the fifth power. 243 will work out if you know the multiplication table.) But in exponential equations, much more often it is not necessary to raise to a power, but vice versa... Find out what number to what degree is hidden behind the number 243, or, say, 343... No calculator will help you here.
You need to know the powers of some numbers by sight, right... Let's practice?
Determine what powers and what numbers the numbers are:
2; 8; 16; 27; 32; 64; 81; 100; 125; 128; 216; 243; 256; 343; 512; 625; 729, 1024.
Answers (in a mess, of course!):
5 4 ; 2 10 ; 7 3 ; 3 5 ; 2 7 ; 10 2 ; 2 6 ; 3 3 ; 2 3 ; 2 1 ; 3 6 ; 2 9 ; 2 8 ; 6 3 ; 5 3 ; 3 4 ; 2 5 ; 4 4 ; 4 2 ; 2 3 ; 9 3 ; 4 5 ; 8 2 ; 4 3 ; 8 3 .
If you look closely, you can see a strange fact. There are significantly more answers than tasks! Well, it happens... For example, 2 6, 4 3, 8 2 - that's all 64.
Let us assume that you have taken note of the information about familiarity with numbers.) Let me also remind you that to solve exponential equations we use all stock of mathematical knowledge. Including those from junior and middle classes. You didn’t go straight to high school, right?)
For example, when solving exponential equations, putting the common factor out of brackets often helps (hello to 7th grade!). Let's look at an example:
3 2x+4 -11 9 x = 210
And again, the first glance is at the foundations! The bases of the degrees are different... Three and nine. But we want them to be the same. Well, in this case the desire is completely fulfilled!) Because:
9 x = (3 2) x = 3 2x
Using the same rules for dealing with degrees:
3 2x+4 = 3 2x ·3 4
That’s great, you can write it down:
3 2x 3 4 - 11 3 2x = 210
We gave an example for the same reasons. So, what is next!? You can't throw out threes... Dead end?
Not at all. Remember the most universal and powerful decision rule everyone math tasks:
If you don’t know what you need, do what you can!
Look, everything will work out).
What's in this exponential equation Can do? Yes, on the left side it just begs to be taken out of brackets! The overall multiplier of 3 2x clearly hints at this. Let's try, and then we'll see:
3 2x (3 4 - 11) = 210
3 4 - 11 = 81 - 11 = 70
The example keeps getting better and better!
We remember that to eliminate grounds we need a pure degree, without any coefficients. The number 70 bothers us. So we divide both sides of the equation by 70, we get:
Oops! Everything got better!
This is the final answer.
It happens, however, that taxiing on the same basis is achieved, but their elimination is not possible. This happens in other types of exponential equations. Let's master this type.
Replacing a variable in solving exponential equations. Examples.
Let's solve the equation:
4 x - 3 2 x +2 = 0
First - as usual. Let's move on to one base. To a deuce.
4 x = (2 2) x = 2 2x
We get the equation:
2 2x - 3 2 x +2 = 0
And this is where we hang out. The previous techniques will not work, no matter how you look at it. We'll have to pull out another powerful and universal method from our arsenal. It's called variable replacement.
The essence of the method is surprisingly simple. Instead of one complex icon (in our case - 2 x) we write another, simpler one (for example - t). Such a seemingly meaningless replacement leads to amazing results!) Everything just becomes clear and understandable!
So let
Then 2 2x = 2 x2 = (2 x) 2 = t 2
In our equation we replace all powers with x's by t:
Well, does it dawn on you?) Have you forgotten the quadratic equations yet? Solving through the discriminant, we get:
The main thing here is not to stop, as happens... This is not the answer yet, we need x, not t. Let's return to the X's, i.e. we make a reverse replacement. First for t 1:
That is,
One root was found. We are looking for the second one from t 2:
Hm... 2 x on the left, 1 on the right... Problem? Not at all! It is enough to remember (from operations with powers, yes...) that a unit is any number to the zero power. Any. Whatever is needed, we will install it. We need a two. Means:
That's it now. We got 2 roots:
This is the answer.
At solving exponential equations at the end sometimes you end up with some kind of awkward expression. Type:
Seven cannot be converted to two through a simple power. They are not relatives... How can we be? Someone may be confused... But the person who read on this site the topic “What is a logarithm?” , just smiles sparingly and writes down with a firm hand the absolutely correct answer:
There cannot be such an answer in tasks “B” on the Unified State Examination. There a specific number is required. But in tasks “C” it’s easy.
This lesson provides examples of solving the most common exponential equations. Let's highlight the main points.
Practical tips:
1. First of all, we look at grounds degrees. We are wondering if it is possible to make them identical. Let's try to do this by actively using actions with degrees. Don't forget that numbers without x's can also be converted to powers!
2. We try to bring the exponential equation to the form when on the left and on the right there are the same numbers in any powers. We use actions with degrees And factorization. What can be counted in numbers, we count.
3. If the second tip doesn’t work, try using variable replacement. The result may be an equation that can be easily solved. Most often - square. Or fractional, which also reduces to square.
4. To successfully solve exponential equations, you need to know the powers of some numbers by sight.
As usual, at the end of the lesson you are invited to decide a little.) On your own. From simple to complex.
Solve exponential equations:
More difficult:
2 x+3 - 2 x+2 - 2 x = 48
9 x - 8 3 x = 9
2 x - 2 0.5x+1 - 8 = 0
Find the product of roots:
2 3's + 2 x = 9
Happened?
Well, then a very complex example (though it can be solved in the mind...):
7 0.13x + 13 0.7x+1 + 2 0.5x+1 = -3
What's more interesting? Then here's a bad example for you. Quite tempting for increased difficulty. Let me hint that in this example, what saves you is ingenuity and the most universal rule for solving all mathematical problems.)
2 5x-1 3 3x-1 5 2x-1 = 720 x
A simpler example, for relaxation):
9 2 x - 4 3 x = 0
And for dessert. Find the sum of the roots of the equation:
x 3 x - 9x + 7 3 x - 63 = 0
Yes Yes! This is a mixed type equation! Which we did not consider in this lesson. Why consider them, they need to be solved!) This lesson is quite enough to solve the equation. Well, you need ingenuity... And may seventh grade help you (this is a hint!).
Answers (in disarray, separated by semicolons):
1; 2; 3; 4; there are no solutions; 2; -2; -5; 4; 0.
Is everything successful? Great.
There is a problem? No problem! Special Section 555 solves all these exponential equations with detailed explanations. What, why, and why. And, of course, there is additional valuable information on working with all sorts of exponential equations. Not just these ones.)
One last fun question to consider. In this lesson we worked with exponential equations. Why didn’t I say a word about ODZ here? In equations, this is a very important thing, by the way...
If you like this site...
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)
You can get acquainted with functions and derivatives.
Equations
How to solve equations?
In this section we will recall (or study, depending on who you choose) the most elementary equations. So what is the equation? In human language, this is some kind of mathematical expression where there is an equal sign and an unknown. Which is usually denoted by the letter "X". Solve the equation- this is to find such values of x that, when substituted into original expression will give us the correct identity. Let me remind you that identity is an expression that is beyond doubt even for a person who is absolutely not burdened with mathematical knowledge. Like 2=2, 0=0, ab=ab, etc. So how to solve equations? Let's figure it out.
There are all sorts of equations (I’m surprised, right?). But all their infinite variety can be divided into only four types.
4. Other.)
All the rest, of course, most of all, yes...) This includes cubic, exponential, logarithmic, trigonometric and all sorts of others. We will work closely with them in the appropriate sections.
I’ll say right away that sometimes the equations of the first three types are so screwed up that you won’t even recognize them... Nothing. We will learn how to unwind them.
And why do we need these four types? And then what linear equations solved in one way square others, fractional rationals - third, A rest They don’t dare at all! Well, it’s not that they can’t decide at all, it’s that I was wrong with mathematics.) It’s just that they have their own special techniques and methods.
But for any (I repeat - for any!) equations provide a reliable and fail-safe basis for solving. Works everywhere and always. This foundation - Sounds scary, but it's very simple. And very (Very!) important.
Actually, the solution to the equation consists of these very transformations. 99% Answer to the question: " How to solve equations?" lies precisely in these transformations. Is the hint clear?)
Identical transformations of equations.
IN any equations To find the unknown, you need to transform and simplify the original example. And so that when the appearance changes the essence of the equation has not changed. Such transformations are called identical or equivalent.
Note that these transformations apply specifically to the equations. There are also identity transformations in mathematics expressions. This is another topic.
Now we will repeat all, all, all basic identical transformations of equations.
Basic because they can be applied to any equations - linear, quadratic, fractional, trigonometric, exponential, logarithmic, etc. and so on.
First identity transformation: you can add (subtract) to both sides of any equation any(but one and the same!) number or expression (including an expression with an unknown!). This does not change the essence of the equation.
By the way, you constantly used this transformation, you just thought that you were transferring some terms from one part of the equation to another with a change of sign. Type:
The case is familiar, we move the two to the right, and we get:
Actually you taken away from both sides of the equation is two. The result is the same:
x+2 - 2 = 3 - 2
Moving terms left and right with a change of sign is simply a shortened version of the first identity transformation. And why do we need such deep knowledge? - you ask. Nothing in the equations. For God's sake, bear it. Just don’t forget to change the sign. But in inequalities, the habit of transference can lead to a dead end...
Second identity transformation: both sides of the equation can be multiplied (divided) by the same thing non-zero number or expression. Here an understandable limitation already appears: multiplying by zero is stupid, and dividing is completely impossible. This is the transformation you use when you solve something cool like
It's clear X= 2. How did you find it? By selection? Or did it just dawn on you? In order not to select and not wait for insight, you need to understand that you are just divided both sides of the equation by 5. When dividing the left side (5x), the five was reduced, leaving pure X. Which is exactly what we needed. And when dividing the right side of (10) by five, the result is, of course, two.
That's all.
It's funny, but these two (only two!) identical transformations are the basis of the solution all equations of mathematics. Wow! It makes sense to look at examples of what and how, right?)
Examples of identical transformations of equations. Main problems.
Let's start with first identity transformation. Transfer left-right.
An example for the younger ones.)
Let's say we need to solve the following equation:
3-2x=5-3x
Let's remember the spell: "with X's - to the left, without X's - to the right!" This spell is instructions for using the first identity transformation.) What expression with an X is on the right? 3x? The answer is incorrect! On our right - 3x! Minus three x! Therefore, when moving to the left, the sign will change to plus. It will turn out:
3-2x+3x=5
So, the X’s were collected in a pile. Let's get into the numbers. There is a three on the left. With what sign? The answer “with none” is not accepted!) In front of the three, indeed, nothing is drawn. And this means that before the three there is plus. So the mathematicians agreed. Nothing is written, which means plus. Therefore, the triple will be transferred to the right side with a minus. We get:
-2x+3x=5-3
There are mere trifles left. On the left - bring similar ones, on the right - count. The answer comes straight away:
In this example, one identity transformation was enough. The second one was not needed. Well, okay.)
An example for older children.)
If you like this site...
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)
You can get acquainted with functions and derivatives.
