How to solve equations with roots

Sometimes in the equations there is a sign of the root. It seems to many students that it is very difficult to solve such equations “with roots” or, more correctly put, irrational equations, but this is not so.

Instruction manual

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Unlike other types of equations, for example, quadratic or linear systems of equations, there is no standard algorithm for solving equations with roots, or more precisely, irrational equations. In each particular case, it is necessary to select the most suitable solution method based on the "appearance" and features of the equation.

The raising of parts of the equation to the same degree.

Most often, to solve equations with roots (irrational equations), the raising of both sides of the equation to the same degree is used. As a rule, to a degree equal to the degree of the root (squared for square root, cube for cubic root). It should be borne in mind that when raising the left and right sides of the equation to an even degree, he may have “extra” roots. Therefore, in this case, one should check the obtained roots by substituting them in the equation. Particular attention in solving equations with square (even) roots should be given to the range of admissible values ​​of the variable (ODZ). Sometimes, the estimation of the ODL alone is enough to solve or significantly simplify the equation.

Example. Solve the equation:

√ (5x-16) = x-2

We square both sides of the equation:

(√ (5x-16)) ² = (x-2) ², whence we successively get:

5x-16 = x²-4x + 4

h²-4x + 4-5x + 16 = 0

h²-9x + 20 = 0

Solving the obtained quadratic equation, we find its roots:

x = (9 ± √ (81-4 ​​* 1 * 20)) / (2 * 1)

x = (9 ± 1) / 2

x1 = 4, x2 = 5

Substituting both found roots into the original equation, we obtain the correct equality. Therefore, both numbers are solutions of the equation.

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Method for introducing a new variable.

Sometimes it’s more convenient to find the roots of an “equation with roots” (an irrational equation) by introducing new variables. In fact, the essence of this method is simply reduced to a more compact record of the solution, i.e. instead of writing a bulky expression each time, it is replaced by a legend.

Example. Solve the equation: 2x + √x-3 = 0

You can solve this equation by squaring both sides. However, the calculations themselves will look rather cumbersome. With the introduction of a new variable, the decision process will turn out to be much more elegant:

We introduce a new variable: y = √ x

Then we get the ordinary quadratic equation:

2y² + y-3 = 0, with variable y.

Solving the resulting equation, we find two roots:

y1 = 1 and y2 = -3 / 2, substituting the found roots in the expression for the new variable (y), we obtain:

√ x = 1 and √ x = -3 / 2.

Since the square root value cannot be a negative number (if you do not touch the area of ​​complex numbers), we get the only solution:

x = 1.

square root solution