Warm-Up Exercises
Find the exact value.
1.
49
2. – 144
ANSWER
7
ANSWER
– 12
3. Use a calculator to approximate the value of 82
16
to the nearest tenth.
ANSWER
2.3
Example 1
Use Properties of Square Roots
Simplify the expression.
a.
2 •
b.
18
18
SOLUTION
9 •
a.
18 =
b.
2•
c.
5
5
5
=
=
9
9
3
10 =
2 = 3 2
20 =
4 •
5 = 2 5
c.
5
9
Example 2
Simplify
Rationalize the Denominator of a Fraction
5
.
2
SOLUTION
5
5
=
2
2
5
2
•
=
2
2
=
10
2
Quotient property of square roots
Multiply by
2
2
Simplify.
.
Checkpoint
Use Properties of Square Roots
Simplify the expression.
1.
12
2.
15 •
3.
7
3
3
ANSWER
2 3
ANSWER
3 5
ANSWER
21
3
Example 3
Solve a Quadratic Equation
Solve x 2 + 1 = 13.
SOLUTION
x 2 + 1 = 13
x 2 = 12
Write original equation.
Subtract 1 from each side.
x = +
– 12
Take the square root of each side.
x = +
– 4• 3
Product property of square roots
x = +
–2 3
Simplify.
ANSWER
The solutions are 2 3 and – 2 3.
Example 3
Solve a Quadratic Equation
CHECK Substitute 2 3 and – 2 3 into the original
equation.
x 2 + 1 = 13
x 2 + 1 = 13
( 2 3)2 + 1 =? 13
( – 2 3)2 + 1 =? 13
?
4 • 3 + 1 = 13
?
4 • 3 + 1 = 13
12 + 1 = 13
12 + 1 = 13
Checkpoint
Solving a Quadratic Equation
Solve the equation.
4. x 2 – 4 = 14
ANSWER
3 2, – 3 2
5. x 2 + 3 = 13
ANSWER
10, – 10
6. 3y 2 = 24
ANSWER
2 2, – 2 2
Example 4
Use a Quadratic Equation as a Model
Skydiving A skydiver jumps from an airplane that is
6000 feet above the ground. The skydiver opens her
parachute when she is 2500 feet above the ground.
a. Write an equation that gives the height (in feet) of
the skydiver above the ground as a function of time
(in seconds).
b. For how many seconds does the skydiver fall before
opening her parachute?
SOLUTION
a. The initial height of the skydiver is h0 = 6000.
h = – 16t 2 + h0
Write falling object model.
Example 4
Use a Quadratic Equation as a Model
h = – 16t 2 + 6000
Substitute 6000 for h0.
b. The height of the skydiver when she opens her
parachute is h = 2500. Substitute 2500 for h in the
model from part (a). Solve for t.
h = – 16t 2 + 6000
2500 = – 16t 2 + 6000
Write model from part (a).
Substitute 2500 for h.
– 3500 = – 16t 2
Subtract 6000 from each side.
– 3500
= t2
– 16
Divide each side by –16.
Example 4
Use a Quadratic Equation as a Model
– 3500
+
=t
– –
16
+
–15 ≈ t
Take the square root of each side.
Use a calculator.
ANSWER
Reject the solution –15, because time must be positive.
The skydiver falls for about 15 seconds before opening
her parachute.
Checkpoint
Use a Quadratic Equation
7. Skydiving A skydiver jumps from a plane that is 5000
feet above the ground. The skydiver opens his
parachute when he is 2000 feet above the ground.
a. Write an equation that gives the height (in feet) of the
skydiver above the ground as a function of time (in
seconds).
ANSWER
h = – 16t 2 + 5000
b. For how many seconds does the skydiver fall before
opening his parachute?
ANSWER
about 14 sec