Essential Question: Describe the procedure for solving a radical equation

A radical equation is an equation that has a variable in a radicand or a variable with a rational exponent. radical equation radical equation not a radical equation

To solve a radical equation, isolate the radical on one side of the equation and then raise both sides of the equation to the inverse power.

3 10x 23( 2) 25x

3 10x

Example: Goal is to get by

itself Subtract 2 from both

sides Inverse of square

root? Square each side Add 2 to both sides Divide both sides by 3

2 3 2 6x 3 2x 3 2 4x

2 2

3 2 4x 3 2 16x 3 18x 6x

YOUR TURN 5 1 6 0x 5 1 6x

2 2

5 1 6x 5 1 36x 5 35x 7x

Example: Goal is to get by itself Divide both sides by 2 Inverse of 2/3 power?

Take each side to 3/2 power

Use absolute values only when taking an even root

Make two equations

Add 2 to both sides

232 2 50x

23( 2)x

232 25x

3

32 23 22 25x

2 125x

2 125 2 125x or x 127 123x or x

Example:

322 3 54x

323 27x

2

3 232 33 27x

3 9x 6x

AssignmentPage 3941 – 12All problemsShow work

Essential Question: Describe the procedure for solving a radical equation

Much like with absolute value equations, you’ll have to check for extraneous solutions.

Part 1 – Solving the problem Get by itself Subtract 5, both sides Square both sides FOIL right side Subtract x & add 3 Factor Solve each

parenthesis

3 5x x 3 5x x

2 2

3 5x x 23 10 25x x x

20 11 28x x 0 ( 4)( 7)x x 4 7x or x

3x

Part 2 – Checking for extraneous solutions Original Problem Two solutions

The only solution is x = 7

3 5x x 4 7x or x x = 4 x = 7

Plug in for all x’s

Simplify under radical

Simplify radical

Combine like terms

Extraneous Real

4 3 5 4 7 3 5 7 1 5 4 4 5 7 1 5 4 2 5 7 6 4 7 7

YOUR TURN

5 1 3x x 5 1 3x x

2 2

5 1 3x x 25 1 6 9x x x

20 11 10x x 0 ( 10)( 1)x x 10 1x or x

Part 2 – Checking for extraneous solutions Original Problem Two solutions

x = 10 x = 1

5 10 131 0 5 11 3 1

10 1 05 3 5 1 3 1 49 3 10 4 3 1 7 13 0 2 13

5 1 3x x 10 1x or x

Real Solution

Extraneous Solution

The only solution is x = 10

Get the two equations on opposite sides

Raise both sides to eliminate the smallest power

FOIL the left sideGet equation = 0

Subtract 3x on both sides Subtract 4 on both sides

Factor the equation

Solve each parenthesis

0.5 0.252 1 3 4 0x x

0.5 0.252 1 3 4x x

4 40.5 0.252 1 3 4x x

22 1 3 4x x 24 4 1 3 4x x x 24 3 0x x

(4 3)( 1) 0x x (4 3) 0 ( 1) 0x or x

34 1x or x

Part 2 – Checking for extraneous solutions Original

Problem Two solutions

Let’s check x = -1

x = 3/4

Plug in for all x’s

Simplify parenthesis

Calculator is your friend

Combine like terms

Real Solution

0.5 0.252 1 3 4 0x x

34 1x or x

0.5 0.253 34 42 1 3 4 0

5 250.5 0

2 4

.250

25 52 2

0.250.50

0.55 52

5

2

0.0

Part 2 – Checking for extraneous solutions Original

Problem Two solutions

¾ is the only real solution

x =-1

Plug in for all x’s

Simplify parenthesis

Extraneous Solution

0.5 0.252 1 3 4 0x x

34 1x or x

0.5 0.252 1 3 4 01 1

0.5 0.251 1 0

(-1)0.5 is not a real number

YOUR TURN

3 2 2 7 0x x 3 2 2 7x x

2 2

3 2 2 7x x 3 2 2 7x x 2 7x 5x

AssignmentPage 394 - 39515 – 29Odd problemsShow work

Note: All problems have at least one real solution. Some will also have one extraneous solution & the book will not give you the extraneous solution, but I want both.