Solving Radical Equations Copyright © 2011 by Lynda Aguirre1

Radical Equations A RADICAL is the symbol best known as a square root symbol. Copyright © 2011 by Lynda Aguirre2 A Radical Equation has both a radical and an equal sign.

Radicals The square root actually has numbers in it that are “understood” and therefore they are not written in. Copyright © 2011 by Lynda Aguirre3 Root = 2 Radicand The object under the radical symbol Power = 1 That means that the Square Root of “Happy Face” is actually… The 2 nd root of “Happy Face” raised to the first power.

Radicals If the Radical is part of an Equation, it can be eliminated by raising both sides to the matching power (i.e. power = root) Copyright © 2011 by Lynda Aguirre4 Root = 2 The 2 nd power cancels out the 2 nd root Since the root is not showing, that makes it a “2” (or 2 nd root) Procedure: Raise both sides to the 2 nd power to eliminate the 2 nd root. Calculate the other side

Radical Equations If the Radical is part of an Equation, it can be eliminated by raising both sides to the matching power (i.e. power = root) Copyright © 2011 by Lynda Aguirre5 Root = 3 The 3 rd power cancels out the 3 rd root If the root is a “3” (or 3 rd root) Procedure: Raise both sides to the 3 rd power to eliminate the 3 rd root. Calculate the other side

Radicals If the Radical is part of an Equation, it can be eliminated by raising both sides to the matching power (i.e. power = root) Copyright © 2011 by Lynda Aguirre6 Root = 2 Since the root is not showing, that makes it a “2” (or 2 nd root) Raise both sides to the 2 nd power to eliminate the 2 nd root. Solve for “x”

Radical Equations If the Radical is part of an Equation, it can be eliminated by raising both sides to the matching power (i.e. power = root) Copyright © 2011 by Lynda Aguirre7 Root = 2 Since the root is not showing, that makes it a “2” (or 2 nd root) Raise both sides to the 2 nd power to eliminate the 2 nd root. Calculate the other side (Use FOIL) Solve for “x” Factor it “2” doesn’t work, so the solution is x = 6 Note: You must check answers when solving radical equations. (Plug the answers into the original problem and see if the sides are equal)

Radical Equations 2) Square both sides 1) Isolate Radical 3) Simplify each side 4) Solve for x SOLUTION Subtract 3 from both sides Subtract 2 from both sides Copyright © 2011 by Lynda Aguirre8

Radical Equations 2) Square both sides 1) Isolate Radical 3) Simplify each side 4) Solve for x SOLUTION Already isolated Subtract 3 from both sides Expand and use FOIL Subtract x from both sides and factor Regular factoring gives us 6 and 1. Note: You must check answers when solving radical equations. (Plug the answers into the original problem and see if the sides are equal) Checking: 1 does not work Copyright © 2011 by Lynda Aguirre9

Radical Equations 2) Square both sides 1) Isolate Radical 3) Simplify each side 4) Solve for x SOLUTION Already isolated Subtract 5 from both sides Expand and use FOIL Subtract x from both sides and factor Regular factoring won’t work, use quadratic formula (see notes) Copyright © 2011 by Lynda Aguirre10

Solving Radical Equations Copyright © 2011 by Lynda Aguirre11 Step 1: Isolate the Radical by moving everything else to the other side of the equation. Move any term that is not under the radical to the other side using Opposite Operations. Step 2: Square both sides If it is a third root, cube both sides, a fourth root? raise to the fourth power, etc. It is a good idea to write parentheses around the terms * If there are two terms remember to use FOIL or one of the formulas* Step 3: Simplify each side Write terms in descending order (highest power to lowest) Distribute, Add or Subtract like terms. Step 4: Solve for x - Linear Equations: Isolate x - Quadratic Equations: Move everything to one side, zero on the other and factor Step 5: Repeat steps 1 through 4 if there is more than one radical in the equation.

Practice 1 doesn’t check Neither solution checks, so the answer is “no solution” Copyright © 2011 by Lynda Aguirre12

For free math notes visit our website: Copyright (c) 2011 by Lynda Greene Aguirre13