How to solve radical equations

A radical equation is an equation that has a variable in a radicand or a variable with a fractional exponent. Follow this guideline to solve radical equations.

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.

$$ Trick: \ If \ \sqrt[n]{y} = c, then \ (\sqrt[n]{y})^n = c^n \ and \ y = c^n $$

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

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

Solving 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

1^0.5 - 1^0.25 = 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