1. 7-5 Solving Square Root and Other Radical Equations

2. Objectives
Solving Radical Equations

3. Vocabulary
If , then and

4. Solving Radical Equations
Solve – x + 1 = –5.
– x + 1 = –5
2x + 1 = 5
Isolate the radical.
( 2x + 1 )2 = 52
Square both sides.
2x + 1 = 25
2x = 24
x = 12
Check: – x + 1 = –5
– (12) –5
– –5
– –5
–5 = –5

5. Solving Radical Equations with Rational Exponents
Solve 3(x + 1) = 24. 3 5
3(x + 1) = 24 3 5
(x + 1) = 8 Divide each side by 3. 3 5 3 5
((x + 1) ) = 8 Raise both sides to the power.
(x + 1)1 = 8 Multiply the exponents and . 5 3
x + 1 = 32 Simplify.
x = 31
Check: 3(x + 1) = 24  
3(31 + 1)
3(25)
3(2)
24 = 24 3 5

6. Real World Example
An artist wants to make a plastic sphere for a sculpture. The plastic weighs 0.8 ounce per cubic inch. The maximum weight of the sphere is to be 80 pounds. The formula for the volume V of a sphere is V = • r 3, where r is the radius of the sphere.
What is the maximum radius the sphere can have? 4 3 4 3
Define: Let r = radius in inches.
Relate: volume of sphere • density of plastic maximum weight
Write:   • r 3 •
< –

7. Continued • 0.8 80 < – r 3 < – r 3 75 < – Use a calculator.•
< –
r 3
3 • 80
4 • • 0.8
< –
r 3 75
< –
Use a calculator. r
< –
The maximum radius is about 2.88 inches.

8. Checking for Extraneous Solutions
Solve x + 2 – 3 = 2x. Check for extraneous solutions.
x + 2 – 3 = 2x
x + 2 = 2x + 3 Isolate the radical.
( x + 2)2 = (2x + 3)2 Square both sides.
0 = 4x2 + 11x + 7 Combine like terms.
0 = (x + 1)(4x + 7) Factor.
x + 2 = 4x2 + 12x + 9 Simplify.
x + 1 = 0 or 4x + 7 = 0 Factor Theorem
x = –1 or x = – 7 4

9. Continued Check: x + 2 – 3 = 2x x + 2 – 3 = 2x
–1 + 2 – (–1) –
1 – 3 – – 3
–2 = –2 – 3 7 4 – 7 4 – 1 4 7 2 – 1 2 7 2 – – 5 2 7 2 – = /
The only solution is –1.

10. Solving Equations with Two Rational Exponents
Solve (x + 1) – (9x + 1) = 0. Check for extraneous solutions. 2 3 1 3
(x + 1) – (9x + 1) = 0 2 3 1
(x + 1) = (9x + 1)
((x + 1) )3 = ((9x + 1) )3 2 3 1
(x + 1)2 = 9x + 1
x2 + 2x + 1 = 9x + 1
x2 – 7x = 0
x(x – 7) = 0
x = 0 or x = 7

11. Continued Check: (x + 1) – (9x + 1) = 0 (x + 1) – (9x + 1) = 0
(0 + 1) – (9(0) + 1) (7 + 1) – (9(7) + 1) 0
(1) – (1) (8) – (64 ) 0
(1) – (12) (8) – (82) 0
1 – 1 = – 8 = 0 2 3 1 3 2 3 1 3 2 3 1 3 2 3 1 3 2 3 1 3 2 3 1 3 2 3 1 3 2 3 1 3 2 3 2 3 2 3 2 3
Both 0 and 7 are solutions.

12. Homework
p 394 # 1, 2, 7, 8, 15, 16