Solve Radical Equation: Step-by-Step Guide
A radical equation has a variable under a square root (or another root). To solve it, get the root by itself on one side and then square both sides to get rid of it. Squaring can sometimes create answers that do not really work (extraneous solutions), so you always have to check your answers in the original equation.
Math Racer skill guide · Algebra 2 · Grade 11
How to solve it, step by step
- Get the root by itself. Move everything else to the other side so the radical is alone.\[\sqrt{f(x)} = g(x)\]
- Square both sides. Square both sides of the equation to remove the square root. (For a cube root, cube both sides.)\[\left(\sqrt{f(x)}\right)^2 = (g(x))^2 \;\Rightarrow\; f(x) = (g(x))^2\]
- Solve the new equation. You usually get a regular equation, like a linear or quadratic. Solve it.\[f(x) - (g(x))^2 = 0\]
- Check for answers that do not work. Put each answer back into the original equation. Throw out any that do not work. These are called extraneous solutions.\[\sqrt{f(x_0)} \stackrel{?}{=} g(x_0)\]
Worked example
Solve sqrt(2x + 7) = x + 2.
Step 1: The root is already by itself: the square root of (2x + 7) = x + 2.
Step 2: Square both sides: 2x + 7 = (x + 2)^2 = x^2 + 4x + 4.
Step 3: Move everything to one side: 0 = x^2 + 4x + 4 - 2x - 7 = x^2 + 2x - 3.
Step 4: Factor: (x + 3)(x - 1) = 0, so x = -3 or x = 1.
Step 5: Check x = 1: the square root of (2 + 7) = the square root of 9 = 3, and 1 + 2 = 3. It works.
Step 6: Check x = -3: the square root of (-6 + 7) = the square root of 1 = 1, but -3 + 2 = -1. Since 1 is not -1, x = -3 does not work.
The only solution is x = 1.
Tips and common mistakes
- Get the root alone before you square. If you square with other terms still there, it gets messy.
- Squaring can create fake answers, so always check.
- When you square something like (x + 2), multiply it out fully: x^2 + 4x + 4, not x^2 + 4.
- If there are two roots, get one alone, square, then get the other root alone and square again.
- Throw out any answer that makes the number under the root negative, or that makes the sides of the equation have opposite signs.
- With cube roots there are no fake answers, but checking is still a good habit.
- A square root cannot be negative, so the other side of the equation can't be negative either.