How to solve radical equations

This lesson will show how to solve radical equations. If an equation has a variable in the radicand or a variable with a fractional exponent, we call the equation a radical equation. 

If the index of the radical is 2, the equation is called square root equation. 

Solving square root equations

To solve square root equations, isolate the radical on one side of the equation. Then, raise both sides of the equation to the same power. First, study the easy example below.

How to solve radical equations
$$ Trick: \ If \ \sqrt[n]{y} = c, then \ (\sqrt[n]{y})^n = c^n \ and \ y = c^n $$
$$ 4 + \sqrt{3x-2} = 6 $$

Isolate the radical on one side of the equation

$$ 4 - 4 + \sqrt{3x-2} = 6 - 4 $$
$$ \sqrt{3x-2} = 2 $$

Since you are dealing with a square root equation, raise both sides of the equation to the second power.

$$ (\sqrt{3x-2})^2 = 2^2 $$

3x - 2 = 4

3x - 2 + 2 = 4 + 2

3x = 6

x = 2

How to solve radical equations with rational exponents.

To solve equations with two radical expressions, isolate one of the radicals.

Solve (2x + 3)0.5 - (11x + 12)0.25 = 0

Isolate one radical

(2x + 3)0.5 - (11x + 12)0.25 + (11x + 12)0.25 = 0 + (11x + 12)0.25

(2x + 3)0.5 = (11x + 12)0.25

Raise both sides to the 4th power

[(2x + 3)0.5]4 = [(11x + 12)0.25]4

(2x + 3)2 = 11x + 12

4x2 + 12x + 9 = 11x + 12

4x2 + 12x - 11x + 9 - 12 = 11x - 11x + 12 - 12

4x2 + x - 3 = 0

(4x - 3)(x + 1) = 0

4x - 3 = 0

x = 3/4

x + 1 = 0

x = -1

x = -1 is a solution.

(2 times -1 + 3)0.5 - (11 times -1 + 12)0.25

10.5 - 10.25 = 1 - 1 = 0

x = 3/4 is also a solution.

(2 times 3/4 + 3)0.5 - (11 times 3/4 + 12)0.25

(6/4 + 3)0.5 - (33/4 + 12)0.25

(4.5)0.5 - (20.25)0.25

Is (4.5)0.5 - (20.25)0.25 = 0

Or is (4.5)0.5 = (20.25)0.25

Raise both sides to the 4th power

[(4.5)0.5]4 = [(20.25)0.25]4

(4.5)2 = 20.25

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