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10.4
Use Square Roots to Solve
Quadratic Equations
Warm Up
Lesson Presentation
Lesson Quiz
10.4 Warm-Up
Evaluate the expression.
1. – √49
ANSWER
–7
2. √200
ANSWER
about 14.1
3. ± √ 121
ANSWER
±11
10.4 Warm-Up
4. A ball is dropped from a height 9 feet above the
ground. How long does it take the ball to hit the
ground?
ANSWER
0.75 sec
10.4 Example 1
Solve the equation.
.
a 2x2 = 8
b. m2 – 18 = – 18
c. b2 + 12 = 5
SOLUTION
a. 2x2 = 8
x2 = 4
x = ± 4 = ± 2
Write original equation.
Divide each side by 2.
Take square roots of each side.
Simplify.
.
ANSWER The solutions are –2 and 2
10.4 Example 1
b. m2 – 18 = – 18
m2 = 0
m= 0
ANSWER
The solution is 0.
Write original equation.
Add 18 to each side.
The square root of 0 is 0.
10.4 Example 1
c. b2 + 12 = 5
b2 = –7
Write original equation.
Subtract 12 from each side.
ANSWER
Negative real numbers do not have real square roots.
So, there is no solution.
10.4 Example 2
Solve 4z2 = 9.
SOLUTION
4z2 = 9
Write original equation.
9
z2 = 4
z=±

Divide each side by 4.
9
4
z=± 3
2
ANSWER
The solutions are – 3 and 3 .
2
2
Take square roots of each side.
Simplify.
10.4 Guided Practice
Solve the equation.
1.
c2 – 25 = 0
ANSWER
–5, 5.
2.
5w2 + 12 = – 8
ANSWER
no solution
3. 2x2 + 11 = 11
ANSWER
0
4. 25x2 = 16
ANSWER
–
5. 9m2 = 100
ANSWER
6. 49b2 + 64 = 0
ANSWER
4
4
5 , 5
– 10 , 10
3
3
no solution
10.4 Example 3
Solve 3x2 – 11 = 7. Round the solutions to the nearest
hundredth.
SOLUTION
3x2 – 11 = 7
3x2 = 18
x2 = 6
x = ± 6
x
± 2.45
Write original equation.
Add 11 to each side.
Divide each side by 3.
Take square roots of each side.
Use a calculator. Round to the nearest
hundredth.
ANSWER
The solutions are about – 2.45 and about 2.45.
10.4 Guided Practice
Solve the equation. Round the solutions to the nearest
hundredth.
7.
x2 + 4 = 14
ANSWER
– 3.16, 3.16
8.
3k2 – 1 = 0
ANSWER
– 0.58, 0.58
9. 2p2 – 7 = 2
ANSWER
– 2.12, 2.12
10.4 Example 4
Solve 6(x – 4)2 = 42. Round the solutions to the nearest
hundredth.
6(x – 4)2 = 42
(x – 4)2 = 7
x – 4 = ± 7
x = 4 ± 7
Write original equation.
Divide each side by 6.
Take square roots of each side.
Add 4 to each side.
ANSWER
The solutions are 4 +  7
6.65 and 4 –  7
1.35.
10.4 Example 4
CHECK
To check the solutions, first write the equation so
that 0 is on one side as follows: 6(x – 4)2 – 42 = 0.
Then graph the related function y = 6(x – 4)2 – 42.
The x-intercepts appear to be about 6.6 and about
1.3. So, each solution checks.
10.4 Guided Practice
Solve the equation. Round the solution to the nearest
hundredth if necessary.
2(x – 2)2 = 18
ANSWER
–1, 5
11. 4(q – 3)2 = 28
ANSWER
0.35, 5.65
12. 3(t + 5)2 = 24
ANSWER
–7.83, –2.17
10.
10.4 Example 5
SPORTS EVENT
During an ice hockey game, a
remote-controlled blimp flies
above the crowd and drops a
numbered table-tennis ball. The
number on the ball corresponds
to a prize. Use the information
in the diagram to find the
amount of time that the ball is in
the air.
10.4 Example 5
SOLUTION
STEP 1
Use the vertical motion model to write an equation for
the height h (in feet) of the ball as a function of time t
(in seconds).
h = –16t2 + vt + s
Vertical motion model
h = –16t2 + 0t + 45
Substitute for v and s.
STEP 2
Find the amount of time the ball is in the air by
substituting 17 for h and solving for t.
10.4 Example 5
h = –16t2 + 45
17 = –16t2 + 45
– 28 = –16t2

Write model.
Substitute 17 for h.
Subtract 45 from each side.
28
= t2
16
Divide each side by 16.
28
=t
16
Take positive square root.
1.32
t
Use a calculator.
ANSWER
The ball is in the air for about 1.32 seconds
10.4 Guided Practice
13. WHAT IF? In Example 5, suppose the table-tennis ball
is released 58 feet above the ground and is caught 12
feet above the ground. Find the amount of time that
the ball is in the air. Round your answer to the nearest
hundredth of a second.
ANSWER
The ball is in the are for about 1.70 second.
10.4 Lesson Quiz
Solve the equation. Round solutions to the nearest
hundredth, if necessary.
1.
4b2 – 13 = 3
ANSWER
2.
9x2 = 25
ANSWER
3.
–2, 2
– 5 , 5
3 3
3n2 –18 = 12
ANSWER
–3.16, 3.16
10.4 Lesson Quiz
4. At a football game you are sitting 32 feet above the
ground. If your hat comes off and falls to the
ground, how long will it be in the air?
ANSWER
About 1.41 sec