11-5 Solving Radical Equations
Preview
Warm Up
California Standards
Lesson Presentation
11-5 Solving Radical Equations
Warm Up
Solve each equation.
1. 3x +5 = 17
4
2. 4x + 1 = 2x – 3
3.
–2
35
4. (x + 7)(x – 4) = 0
–7, 4
5. x2 – 11x + 30 = 0
6, 5
6. x2 = 2x + 15
5, –3
11-5 Solving Radical Equations
California
Standards
Extension of
2.0 Students understand
and use such operations as taking the opposite,
finding the reciprocal, taking a root, and raising
to a fractional power. They understand and use
the rules of exponents.
11-5 Solving Radical Equations
Vocabulary
radical equation
11-5 Solving Radical Equations
A radical equation is an equation that contains
a variable within a radical. In this chapter, you
will study radical equations that contain only
square roots.
Recall that you use inverse operations to solve
equations. For nonnegative numbers, squaring
and taking the square root are inverse operations.
When an equation contains a variable within a
square root, you can solve by squaring both sides
of the equation.
11-5 Solving Radical Equations
=
11-5 Solving Radical Equations
Additional Example 1A: Solving Simple Radical
Equations
Solve the equation. Check your answer.
Square both sides.
x = 25
Check
5
5
Substitute 25 for x in the original
equation.
5
Simplify.
11-5 Solving Radical Equations
Additional Example 1B: Solving Simple Radical
Equations
Solve the equation. Check your answer.
Square both sides.
100 = 2x
50 = x
Divide both sides by 2.
Check
Substitute 50 for x in the original
equation.
10
10
Simplify.
11-5 Solving Radical Equations
Check It Out! Example 1a
Solve the equation. Check your answer.
Square both sides.
Simplify.
Check
6
6
Substitute 36 for x in the original
equation.
Simplify.
11-5 Solving Radical Equations
Check It Out! Example 1b
Solve the equation. Check your answer.
Square both sides.
81 = 27x
3=x
Divide both sides by 27.
Check
Substitute 3 for x in the original
equation.

Simplify.
11-5 Solving Radical Equations
Check It Out! Example 1c
Solve the equation. Check your answer.
Square both sides.
3x = 1
Divide both sides by 3.
Check
Substitute for x in the original
equation.

Simplify.
11-5 Solving Radical Equations
Check It Out! Example 1d
Solve the equation. Check your answer.
Square both sides.
x = 12
Multiply both sides by 3.
11-5 Solving Radical Equations
Check It Out! Example 1d continued
Solve the equation. Check your answer.
Check
Substitute 12 for x.
Simplify.
2
2
11-5 Solving Radical Equations
Some square-root equations do not have the
square root isolated. To solve these equations,
you may have to isolate the square root before
squaring both sides. You can do this by using
one or more inverse operations.
11-5 Solving Radical Equations
Additional Example 2A: Solving Simple Radical
Equations
Solve the equation. Check your answer.
Add 4 to both sides.
Square both sides.
x = 81
Check
9–4 5
5 5
11-5 Solving Radical Equations
Additional Example 2B: Solving Simple Radical
Equations
Solve the equation. Check your answer.
Square both sides.
x = 46
Subtract 3 from both sides.
Check
7
7
11-5 Solving Radical Equations
Additional Example 2C: Solving Simple Radical
Equations
Solve the equation. Check your answer.
Subtract 6 from both sides.
Square both sides.
5x + 1 = 16
5x = 15
x=3
Subtract 1 from both sides.
Divide both sides by 5.
11-5 Solving Radical Equations
Additional Example 2C Continued
Solve the equation. Check your answer.
Check
4+6
10
10
10

11-5 Solving Radical Equations
Check It Out! Example 2a
Solve the equation. Check your answer.
Add 2 to both sides.
Square both sides.
x=9
Check
1
1
11-5 Solving Radical Equations
Check It Out! Example 2b
Solve the equation. Check your answer.
Square both sides.
x = 18
Subtract 7 from both sides.
Check
5
5
11-5 Solving Radical Equations
Check It Out! Example 2c
Solve the equation. Check your answer.
Add 1 to both sides.
Square both sides.
3x = 9
x=3
Subtract 7 from both sides.
Divide both sides by 3.
11-5 Solving Radical Equations
Check It Out! Example 2c Continued
Solve the equation. Check your answer.
Check
3
3
11-5 Solving Radical Equations
Additional Example 3A: Solving Radical Equations
by Multiplying or Dividing
Solve the equation. Check your answer.
Method 1
Divide both sides by 4.
Square both sides.
x = 64
11-5 Solving Radical Equations
Additional Example 3A Continued
Solve the equation. Check your answer.
Method 2
Square both sides.
x = 64
Divide both sides by 16.
11-5 Solving Radical Equations
Additional Example 3A Continued
Solve the equation. Check your answer.
Check
Substitute 64 for x in the
original equation.
32
32 
Simplify.
11-5 Solving Radical Equations
Additional Example 3B: Solving Radical Equations
by Multiplying or Dividing
Solve the equation. Check your answer.
Method 1
Multiply both sides by 2.
Square both sides.
144 = x
11-5 Solving Radical Equations
Additional Example 3B Continued
Solve the equation. Check your answer.
Method 2
Square both sides.
Multiply both sides by 4.
144 = x
11-5 Solving Radical Equations
Additional Example 3B Continued
Solve the equation. Check your answer.
Check
Substitute 144 for x in the
original equation.
Simplify.
6
6

11-5 Solving Radical Equations
Check It Out! Example 3a
Solve the equation. Check your answer.
Method 1
Divide both sides by 2.
Square both sides.
11-5 Solving Radical Equations
Check It Out! Example 3a Continued
Solve the equation. Check your answer.
Method 2
Square both sides.
x = 121
Divide both sides by 4.
11-5 Solving Radical Equations
Check It Out! Example 3a Continued
Solve the equation. Check your answer.
Check
Substitute 121 for x in the
original equation.

Simplify.
11-5 Solving Radical Equations
Check It Out! Example 3b
Solve the equation. Check your answer.
Method 1
Multiply both sides by 4.
Square both sides.
64 = x
11-5 Solving Radical Equations
Check It Out! Example 3b Continued
Solve the equation. Check your answer.
Method 2
Square both sides.
Multiply both sides by 16.
11-5 Solving Radical Equations
Check It Out! Example 3b Continued
Solve the equation. Check your answer.
Check
Substitute 64 for x in the
original equation.
Simplify.

11-5 Solving Radical Equations
Check It Out! Example 3c
Solve the equation. Check your answer.
Method 1
Multiply both sides by 5.
Square both sides.
Divide both sides by 4.
x = 100
11-5 Solving Radical Equations
Check It Out! Example 3c Continued
Solve the equation. Check your answer.
Method 2
Square both sides.
4x = 400
x = 100
Multiply both sides by 25.
Divide both sides by 4.
11-5 Solving Radical Equations
Check It Out! Example 3c Continued
Solve the equation. Check your answer.
Check
4
4
Substitute 100 for x in the
original equation.
4
Simplify.

11-5 Solving Radical Equations
Additional Example 4A: Solving Radical Equations
with Square Roots on Both Sides
Solve the equation. Check your answer.
Square both sides.
2x – 1 = x + 7
x=8
Check
Add 1 to both sides and
subtract x from both sides.

11-5 Solving Radical Equations
Additional Example 4B: Solving Radical Equations
with Square Roots on Both Sides
Solve the equation. Check your answer.
Add
to both sides.
Square both sides.
5x – 4 = 6
5x = 10
x=2
Add 4 to both sides.
Divide both sides by 5.
11-5 Solving Radical Equations
Additional Example 4B Continued
Solve the equation. Check your answer.
Check
0
0
11-5 Solving Radical Equations
Check It Out! Example 4a
Solve the equation. Check your answer.
Square both sides.
2x = 4
x=2
Subtract x from both sides and
subtract 2 from both sides.
Divide both sides by 2.
11-5 Solving Radical Equations
Check It Out! Example 4a Continued
Solve the equation. Check your answer.
Check

11-5 Solving Radical Equations
Check It Out! Example 4b
Solve the equation. Check your answer.
Add
to both sides.
Square both sides.
2x – 5 = 6
2x = 11
Add 5 to both sides.
Divide both sides by 2.
11-5 Solving Radical Equations
Check It Out! Example 4b Continued
Solve the equation. Check your answer.
Check
0 0
11-5 Solving Radical Equations
Squaring both sides of an equation may result in an
extraneous solution.
Suppose your original
equation is x = 3.
Square both sides. Now you
have a new equation.
Solve this new equation for x
by taking the square root of
both sides.
x=3
x2 = 9
x = 3 or x = –3
11-5 Solving Radical Equations
Now there are two solutions of the new
equation. One (x = 3) is the original equation.
The other (x = –3) is extraneous–it is not a
solution of the original equation. Because of
extraneous solutions, it is especially important
to check your answers to radical equations.
11-5 Solving Radical Equations
Additional Example 5A: Extraneous Solutions
Solve
Check your answer.
Subtract 12 from each sides.
Square both sides
6x = 36
x=6
Divide both sides by 6.
11-5 Solving Radical Equations
Additional Example 5A Continued
Solve
Check your answer.
Check
Substitute 6 for x in the
equation.
18
6 does not check; Ø.
6

11-5 Solving Radical Equations
Additional Example 5B: Extraneous Solutions
Solve
Check your answer.
Square both sides
x2 = 2x + 3
x2 – 2x – 3 = 0
(x – 3)(x + 1) = 0
x – 3 = 0 or x + 1 = 0
x = 3 or
x = –1
Write in standard form.
Factor.
Zero-Product Property
Solve for x.
11-5 Solving Radical Equations
Additional Example 5B Continued
Solve
Check your answer.
Check
Substitute –1 for x in the
equation.
–1
1
Substitute 3 for x in the
equation.
3
3
–1 does not check; it is extraneous. The only
solution is 3.
11-5 Solving Radical Equations
Check It Out! Example 5a
Solve the equation. Check your answer.
Subtract 11 from both sides.
Square both sides.
x=5
Simplify.
11-5 Solving Radical Equations
Check It Out! Example 5a Continued
Solve the equation. Check your answer.
Check
Substitute 5 for x in the
equation.
16
6
The answer is extraneous.
11-5 Solving Radical Equations
Check It Out! Example 5b
Solve the equation. Check your answer.
Square both sides
x2 = –3x – 2
x2 + 3x + 2 = 0
(x + 1)(x + 2) = 0
x + 1 = 0 or x + 2 = 0
x = –1 or x = –2
Write in standard form.
Factor.
Zero-Product Property
Solve for x.
11-5 Solving Radical Equations
Check It Out! Example 5b Continued
Solve the equation. Check your answer.
Check

–2
2
Substitute –1 for x in the
equation.
Substitute –2 for x in the
equation.
Both answers are extraneous.
11-5 Solving Radical Equations
Check It Out! Example 5c
Solve the equation. Check your answer.
Square both sides.
x2 – 4x + 4 = x
Subtract x from both sides.
x2 – 5x + 4 = 0
(x – 1)(x – 4) = 0
x – 1 = 0 or x – 4 = 0
x = 1 or x = 4
Factor.
Zero-Product Property
Solve for x.
11-5 Solving Radical Equations
Check It Out! Example 5c Continued
Solve the equation. Check your answer.
Check

2
2
Substitute 1 for x in the
equation.
Substitute 4 for x in the
equation.
1 does not check; it is extraneous. The only
solution is 4.
11-5 Solving Radical Equations
Additional Example 6: Geometry Application
A triangle has an area of 36
square feet, its base is 8 feet,
and its height is
feet. What
is the value of x? What is the
height of the triangle?
8 ft
Use the formula for area of a triangle.
Substitute 8 for b, 36 for A, and
for h.
Simplify.
Divide both sides by 4.
11-5 Solving Radical Equations
Additional Example 6 Continued
A triangle has an area of 36
square feet, its base is 8 feet,
and its height is
feet. What
is the value of x? What is the
height of the triangle?
Square both sides.
81 = x – 1
82 = x
8 ft
11-5 Solving Radical Equations
Additional Example 6 Continued
A triangle has an area of 36
square feet, its base is 8 feet,
and its height is
feet. What
is the value of x? What is the
height of the triangle?
8 ft
Check
Substitute 82 for x.
36
36 
The value of x is 82. The height of the triangle
is
9 feet.
11-5 Solving Radical Equations
Check It Out! Example 6
A rectangle has an area of 15
cm2. Its width is 5 cm, and
its length is (
) cm. What
is the value of x? What is the
length of the rectangle?
A = lw
5
Use the formula for area of a rectangle.
Substitute 5 for w, 15 for A, and
for l.
Divide both sides by 5.
11-5 Solving Radical Equations
Check It Out! Example 6 Continued
A rectangle has an area of 15
cm2. Its width is 5 cm, and
its length is (
) cm. What
is the value of x? What is the
length of the rectangle?
5
Square both sides.
8=x
The value of x is 8. The length of the rectangle is
cm.
11-5 Solving Radical Equations
Check It Out! Example 6 Continued
A rectangle has an area of 15
cm2. Its width is 5 cm, and
its length is (
) cm. What
is the value of x? What is the
length of the rectangle?
Check
A = lw
Substitute 8 for x.
15
15 
5
11-5 Solving Radical Equations
Lesson Quiz
Solve each equation. Check your answer.
1.
2.
36
3.
4.
ø
5.
45
4
6.
11
4
7. A triangle has an area of 48 square feet, its base
is 6 feet, and its height is
feet. What is the
value of x? What is the height of the triangle?
253; 16 ft