solving radical equations copyright © 2011 by lynda aguirre1

Copyright © 2011 by Lynda Aguirre 1

Solving Radical Equations


Radical EquationsA RADICAL is the symbol best known as a

square root symbol.

√❑A Radical Equation has both a radical

and an equal sign.

= 3


RadicalsThe square root actually has numbers in it that are “understood”

and therefore they are not written in.

2√❑1Root = 2

RadicandThe object under the

radical symbol

Power = 1

√❑That means that the Square Root of “Happy Face” is actually…

The 2nd root of “Happy Face” raised to the first power.


RadicalsIf the Radical is part of an Equation, it can be eliminated by raising both sides to the matching power (i.e. power = root)

Root = 2

The 2nd power cancels out the

2nd root

Since the root is not showing, that makes it a “2” (or 2nd root)

Procedure: Raise both sides to the 2nd power to eliminate the 2nd root.

= 3

= Calculate the other side

9


Radical EquationsIf the Radical is part of an Equation, it can be eliminated by raising both sides to the matching power (i.e. power = root)

Root = 3

The 3rd power cancels out the

3rd root

If the root is a “3” (or 3rd root)

Procedure: Raise both sides to the 3rd power to eliminate the 3rd root.

= 4

= Calculate the other side

64


RadicalsIf the Radical is part of an Equation, it can be eliminated by raising both sides to the matching power (i.e. power = root)

Root = 2Since the root is not showing, that makes it a “2” (or 2nd root)

Raise both sides to the 2nd power to eliminate the 2nd root.

= 5

=

Solve for “x”


Radical EquationsIf the Radical is part of an Equation, it can be eliminated by raising both sides to the matching power (i.e. power = root)

Root = 2

Since the root is not showing, that makes it a “2” (or 2nd root)

Raise both sides to the 2nd power to eliminate the 2nd root.√2 𝑥−3=𝑥−3

= Calculate the other side(Use FOIL)

Solve for “x”

0 Factor it

(𝑥−2 ) (𝑥−6 )=0“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



Radical Equations


1) Isolate Radical


4) Solve for x

SOLUTION

Already isolated


3

(√𝑥+3 )2=(𝑥−3 )2

𝑥+3=(𝑥−3)(𝑥−3)

Expand and use FOIL

9−3−3

0 6 Subtract x from both sides and

factor

−𝑥−𝑥

0Regular factoring gives

us 6 and 1. x =

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


Radical Equations


1) Isolate Radical


4) Solve for x

SOLUTION

Already isolated


√𝑥+5=𝑥−2

(√𝑥+5 )2=(𝑥−2 )2

𝑥+5=(𝑥−2)(𝑥−2)

Expand and use FOIL

𝑥+5=𝑥2−4 𝑥+4−5−5

𝑥=𝑥2−4 𝑥−1 Subtract x from both sides and

factor−𝑥−𝑥

0 Regular factoring won’t work, use quadratic formula (see notes)

𝑥=5±√295


Solving Radical EquationsStep 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

√𝑥−3=8 𝑥=121

√ 4 𝑥=𝑥−3 𝑥=1 ,9 1 doesn’t check

3√𝑥=7 𝑥=73𝑜𝑟 343

√𝑥+2=3 𝑥+2 𝑥=2 ,9

Neither solution checks, so the answer is “no

solution”

Copyright (c) 2011 by Lynda Greene Aguirre 13

For free math notes visit our website:

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solving radical equations copyright © 2011 by lynda aguirre1

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