Download 11-9 Solving Radical Equations

11-9
Solving Radical Equations
Warm Up
Solve each equation.
1. 3x +5 = 17
2. 4x + 1 = 2x – 3
3.
4. (x + 7)(x – 4) = 0
5. x2 – 11x + 30 = 0
6. x2 = 2x + 15
11-9
Solving Radical Equations
Learning Goal
1.  Students will be able to solve radical equations
2.  Students will be able to solve one-step radical
equations using addition and subtraction
3.  Students will be able to solve multi-step
radical equations using multiplication and
division
4.  Students will be able to solve radical equations
with multiple radicals
Solving Radical Equations
11-9
A radical equation is an equation that contains
a variable within a radical.
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, square both sides of the equation to
solve.
4 2 = 16
25 = 5
(4 ) =
2
(
25
)
2
16
= ( 5)
2
2
4
( )=4
4=4
25 = 25
2
5
( )=5
11-9
Solving Radical Equations
11-9
Solving Radical Equations
Example 1: Solving Simple Radical Equations
Solve the equation. Check your answer.
A.
x =5
B. 10 = 2x
11-9
Solving Radical Equations
Example 1: Solving Simple Radical Equations
Solve the equation. Check your answer.
a.
x =6
b. 9 = 27x
c.
d.
x =7
e.
b. 9 = 27x
20a = 10
3x = 1
11-9
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.
11-9
Solving Radical Equations
Example 2: Solving Simple Radical Equations
Solve the equation. Check your answer.
A.
x −4=5
B.
x+3 = 7
C.
5x +1 + 6 = 10
11-9
Solving Radical Equations
Example 2: Solving Simple Radical Equations
Solve the equation. Check your answer.
a.
x − 2 =1
b.
x+7 =5
d.
x + 6 = 11
e.
2−a =3
c.
3x + 7 −1 = 3
11-9
Solving Radical Equations
Example 3: Solving Radical Equations by Multiplying
or Dividing
Solve the equation. Check your answer.
A. 4 x = 32
x
B. 6 =
2
11-9
Solving Radical Equations
Example 3: Solving Radical Equations by Multiplying
or Dividing
Solve the equation. Check your answer.
a. 2 x = 22
x
b. 2 =
4
5 x
= 10
d.
6
e. 2 x = 8
g. 3 x = 1
h. 13 2x = 26
2
j.
x−7
=1
3
k. 4 2x −1 = 12
2 x
=4
c.
5
f.
x
=3
3
i.
x
=2
5
11-9
Solving Radical Equations
Example 4: Solving Radical Equations with Square
Roots on Both Sides
Solve the equation. Check your answer.
A.
2x −1 = x + 7
B.
5x − 4 − 6 = 0
11-9
Solving Radical Equations
Example 4: Solving Radical Equations with Square
Roots on Both Sides
Solve the equation. Check your answer.
a.
3x + 2 = x + 6
b.
c.
5 − x = 6x − 2
d. 0 = 2x − x + 3
e.
−x = 2x +1
f.
2x − 5 = 6 = 0
x−5+5= 0
11-9
Solving Radical Equations
Squaring both sides of an equation may result in an
extraneous solution—a number that is not a
solution of the original equation.
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
11-9
Solving Radical Equations
Now there are two solutions. 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
important to check your answers.
11-9
Solving Radical Equations
Example 5A: Extraneous Solutions
Solve. Check your answer.
B.
A.
11-9
a.
Solving Radical Equations
Check It Out! Example 5a
Solve the equation. Check your answer.
c.
b.
11-9
Solving Radical Equations
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
11-9
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?
5
11-9
Solving Radical Equations
Lesson Quiz: Part I
Solve each equation. Check your answer.
1.
2.
3.
4.
5.
6.
11-9
Solving Radical Equations
Lesson Quiz: Part II
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