How to find the root of the equation ?

How to find the root of the equation ?

One of the major branches of mathematics is a section devoted to the solution of equations and finding roots of equations.

Before you find the root of the equation, one must first understand what it is.

root of the equation - the value of the unknown quantity in the equation, denoted by letters (often - x, y, but there may be other letters).This was discussed in our article - What is the root of the equation.

consider how to find all the roots, to the different types of equations and concrete examples.

equation of the form ax + b = 0

This is a linear equation with one variable, where a and b - number of, x-root of the equation.

number of roots of the equation depends on the values ​​of a and b:

  1. If a = b = 0, then the equation has an infinite number of roots.
  2. If a = 0, b is not 0, then the equation has no roots.
  3. If not equal to 0, then the root is given by: x = - (b / a)

Example:

  • 5x + 2 = 0
  • a = 5, b = 2
  • x = - (2/5)
  • x = -0,4

Answer: root of the equation is equal to 0,4

equation of the form ax² + bx + c = 0.

This is a quadratic equation.There are several ways of finding the roots of a quadratic equation in.We consider the general, which is suitable for solutions for all values ​​of a, b and c.

First you need to find the value of the discriminant (D) of this equation.

For this purpose there is a formula:

  • D = b2-4ac

Depending on which to learn discriminant, there are 3 options for further solutions:

  1. If D & gt; 0, the roots of 2. And they are calculated on theformulas:
    • x1 = (-b + √ D) / 2a.
    • x2 = (-b - √ D) / 2a
  2. If D = 0, the root of one - it can be found by the formula: x = - (b / 2a)
  3. If D & lt; 0, the equation does notIt has roots.

Example:

  • x2 + 3x-4 = 0

where a = 1, b = 3, p = -4

  • D = 32 - (4 * 1 * (- 4))
  • D = 9- (-16)
  • D = 9 + 16
  • D = 25

D & gt; 0, then in the equation 2 is the root.

  • √D = √25 = 5

substitute all the values ​​in our formula:

  • x1 = (-3 +5) / 2 * 1
  • x1 = 2/2
  • x1 = 1, x2
  • = (-3-5) / 2 * 1
  • x2 = (-8) / 2
  • x2 = -4

A: The roots of the equation are 1 and -4.

equation of the form ax3 + bx2 + cx + d = 0

This is a cubic equation.

There are special math formula Cardano, which can solve this equation, but they are very complicated.We'll go to other, more intuitive way.

Cubic equations always have at least one root, and its value is usually an integer from -3 to 3. That is, we have equations will in turn substitute for x numbers: -3, -2, -1, 0, 1,2 and 3. It is X1.

It is much easier and faster than you think, and certainly easier than the formulas of Cardano.

After we find x1, we go to the search X2 and X3.

To do this, divide our equation in the (x-x1) - this can be done by making the brackets.We must stay the quadratic equation, which we solve a little earlier in this article.

Example:

  • x3 - 3x2 - 13x + 15 = 0

selection method we find that X1 = 1, that is, we have to divide our equation in the (x-1)

As a result, we get:

  • x2 - 2x - 15 = 0

we got a quadratic equation.We solve it as described above.And we come to the fact that it has two roots: - 3 and 5.

Answer:

  • roots of the equation: x1 = 1, x2 = -3, x 3 = 5.

Even more information can be found in the article How to solve the roots.