Course: Intermediate Algebra
Topic: Nonlinear Equations and Applications
Subtopic: Radical Equations


Today we will transition from radical expressions to radical equations. We will now be able to solve equations that contain a radical (or two!) for x. The process involves eliminating the radical and solving the resulting equation. You eliminate the radical by isolating it and squaring (or cubing, fourthing, etc.) both sides of the equation.

CAUTION: Be sure to square (or cube, etc.) the entire side of the equation not individual terms. I.e., squaring both sides of √(x) = √(y)+2 does not make x=y+4. Actually it makes x = (√(y)+2)2 = (√(y)+2)(√(y)+2) which must then by FOILed.

CAUTION: You must always check your answers when you even-power both sides due to the potential for extraneous solutions. Check by plugging each answer back into the original equation. Then simplify each side to see if they match (and don't power each side to eliminate the radical as that is where the potential error can enter in the first place!).

CAUTION: Recall that an even root can never be equal to a negative number, so something like √(x-5) = -3 is automatically no solution. Don't waste your time trying to solve it.

You should be able to solve more complicated radical equations graphically. See the supplemental resources below for assistance.


By the end of this topic you should know and be prepared to be tested on:


Terms you should be able to define: radical equation, extraneous solution

Supplemental Resources

To solve a radical equation using an electronic grapher you must be able to graph each side of the original equation as a separate function and then find the intersection point (using the INTERSECTION feature of your grapher). If you have a handheld graphing calculator then these sites may help: