Here are the solutions to the warm-up equations: 1) 2x^2 - 32 = 0 2x^2 = 32 x^2 = 16 x = ±4 2) 2w^2 + 13 = 11 2w^2 = -2 w^2 = -1 No real solution 3) 25b^2 + 11 = 15 25b^2 = 4 b^2 = 1/6 b = ±1/5 4) x^2 + 11 = 24 x^2 = 13 x = ±√13 5) (x - 7)^2 = 6 (x - 7) = ±√

2. Solving x = d 2
 To use square roots to solve a quadratic equation of the
form ax 2 + c = 0 , first isolate x 2 on one side of the
equation to obtain x 2 = d . Then use the following:
 If d > 0, then x 2 = d has two solutions: x = ± d
 If d = 0, then x = d has one solution: x = 0
2
 If d < 0, then x = d has no solution.
2

3. Example 1 Solve quadratic equations
Solve the equation.
a. 2x2 = 8 b. m2 – 18 = –18 c. b2 + 12 = 5
a. 2x2 = 8 Write original equation.
x2 = 4 Divide each side by 2.
x = + 4
– Take square roots of each side.
x = +2
– Simplify.
ANSWER The solutions are –2 and 2.

4. Example 1 Solve quadratic equations
b. m2 – 18 = –18 Write original equation.
m2 = 0 Add 18 to each side.
m = 0 The square root of 0 is 0.
ANSWER The solution is 0.
c. b2 + 12 = 5 Write original equation.
b2 = –7 Subtract 12 from each side.
ANSWER
Negative real numbers do not have real square roots.
So, there is no solution.

5. Example 2 Take square roots of a fraction
Solve 4z2 = 9.
SOLUTION
4z2 = 9 Write original equation.
9
z2 = Divide each side by 4.
4
9
z = +
– Take square roots of each side.
4
3
z = +
– Simplify.
2

6. Example 2 Take square roots of a fraction
3 3
ANSWER The solutions are – and .
2 2

7. Example 3 Solve a quadratic equation
Solve 3x2 – 11 = 13.
SOLUTION
3x2 – 11 = 13 Write original equation.
3x2 = 24 Add 11 to each side.
x2 = 8 Divide each side by 3.
x = +
– 8 Take square roots of each side.
x = +2
– 2 Simplify.
ANSWER The solutions are –2 2 and 2 2.

8. Example 4 Solve a quadratic equation
Solve 6 ( x – 4 )2 = 42.
6 ( x – 4 )2 = 42 Write original equation.
( x – 4 )2 = 7 Divide each side by 6.
x – 4 = +
– 7 Take square roots of each side.
x = 4 –
+ 7 Add 4 to each side.
ANSWER The solutions are 4 + 7 and 4 – 7.

9. Example 5 Solve a multi-step problem
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. Example 5 Solve a multi-step problem
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 the time
t (in seconds) after it is dropped.
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.

11. Example 5 Solve a multi-step problem
17 = – 16t2 + 45 Substitute 17 for h in model.
– 28 = – 16t2 Subtract 45 from each side.
28
= t2 Divide each side by –16.
16
28 7
= = t Take positive square root.
16 2
1.32 ≈ t Use a calculator.
ANSWER The ball is in the air for about 1.32 seconds.

12. 10.5 Warm-Up
Solve the equation.
1. 2x 2 - 32 = 0
2. 2w2 +13 =11
3. 25b2 +11 =15
4. x 2 +11= 24
( x - 7) =6
2
5.