1. 1. Write 15x2 + 6x = 14x2 – 12 in standard form.
Lesson 4.8, For use with pages
1. Write 15x2 + 6x = 14x2 – 12 in standard form.
ANSWER
x2 + 6x +12 = 0
2. Evaluate b2 – 4ac when a = 3, b = –6, and c = 5.
ANSWER
–24

2. 3. A student is solving an equation by completing the
Lesson 4.8, For use with pages
3. A student is solving an equation by completing the
square. Write the step in the solution that appears
just before “(x – 3) = 5.” + _
ANSWER
(x – 3)2 = 25

3. Questions 4.7

6. Write original equation.
EXAMPLE 1
Solve an equation with two real solutions
Solve x2 + 3x = 2.
x2 + 3x = 2
Write original equation.
x2 + 3x – 2 = 0
Write in standard form.
x =
– b b2 – 4ac 2a
Quadratic formula
x =
– – 4(1)(–2)
2(1)
a = 1, b = 3, c = –2
x =
– 3 + 17 2
Simplify.
The solutions are
x =
– 3 + 17 2
0.56 and
– 3 –
– 3.56.
ANSWER

7. EXAMPLE 1
Solve an equation with two real solutions
CHECK
Graph y = x2 + 3x – 2 and note that the x-intercepts are about 0.56 and about – 3.56. 

8. Write original equation.
EXAMPLE 2
Solve an equation with one real solutions
Solve 25x2 – 18x = 12x – 9.
25x2 – 18x = 12x – 9.
Write original equation.
25x2 – 30x + 9 = 0.
Write in standard form.
x =
(–30)2 – 4(25)(9)
2(25)
a = 25, b = –30, c = 9
x = 50
Simplify.
x = 3 5
Simplify. 3 5
The solution is
ANSWER

9. EXAMPLE 2
Solve an equation with one real solutions
CHECK
Graph y = –5x2 – 30x + 9 and note that the only x-intercept is 0.6 = . 3  5

10. Write original equation.
EXAMPLE 3
Solve an equation with imaginary solutions
Solve –x2 + 4x = 5.
–x2 + 4x = 5
Write original equation.
–x2 + 4x – 5 = 0.
Write in standard form.
x =
– 42– 4(– 1)(– 5)
2(– 1)
a = –1, b = 4, c = –5
x =
– – 4
– 2
Simplify.
– 4+ 2i
x =
– 2
Rewrite using the imaginary unit i.
x = 2 + i
Simplify.
The solution is 2 + i and 2 – i.
ANSWER

11. EXAMPLE 3
Solve an equation with imaginary solutions
CHECK
Graph y = 2x2 + 4x – 5. There are no x-intercepts. So, the original equation has no real solutions. The algebraic check for the imaginary solution 2 + i is shown.
–(2 + i)2 + 4(2 + i) = 5 ?
–3 – 4i i = 5 ?
5 = 5 

12. Write original equation.
GUIDED PRACTICE
for Examples 1, 2, and 3
Use the quadratic formula to solve the equation.
x2 = 6x – 4
SOLUTION
x2 = 6x – 4
Write original equation.
x2 – 6x + 4 = 0
Write in standard form.
x =
– b b2 – 4ac 2a
Quadratic formula
x =
– (– 6) (– 6)2 – 4(1)(4)
2(1)
a = 1, b = – 6, c = 4
x =
+ 3 + 20 2
Simplify.

13. GUIDED PRACTICE
for Examples 1, 2, and 3
ANSWER
The solutions are
x =
3 + 20 2
and
3 –
= 3 – =

14. Write original equation.
GUIDED PRACTICE
for Examples 1, 2, and 3
Use the quadratic formula to solve the equation.
4x2 – 10x = 2x – 9
SOLUTION
4x2 – 10x = 2x – 9
Write original equation.
4x2 – 12x + 9 = 0
Write in standard form.
x =
– b b2 – 4ac 2a
Quadratic formula
x =
– (– 12) (– 12)2 – 4(4)(9)
2(4)
a = 4, b = – 12, c = 9
x =
12 + 8
Simplify.

15. GUIDED PRACTICE
for Examples 1, 2, and 3
ANSWER 3 2
The solution is = 1

16. Write original equation.
GUIDED PRACTICE
for Examples 1, 2, and 3
Use the quadratic formula to solve the equation.
7x – 5x2 – 4 = 2x + 3
SOLUTION
7x – 5x2 – 4 = 2x + 3
Write original equation.
– 5x2 + 5x – 7 = 0
Write in standard form.
x =
– b b2 – 4ac 2a
Quadratic formula
x =
– (5) (5)2 – 4(– 5)(–7)
2(– 5)
a = – 5, b = 5, c = – 7
–115
x =
– 5 +
– 10
Simplify.

17. Rewrite using the imaginary unit i.
GUIDED PRACTICE
for Examples 1, 2, and 3
115
x =
– 5 + i
– 10
Rewrite using the imaginary unit i.
115
x =
5 + i 10
Simplify.
ANSWER
The solutions are
and
115
5 + i 10
5 – i

18. EXAMPLE 4
Solve a quadratic inequality using a table
Solve x2 + x ≤ 6 using a table.
SOLUTION
Rewrite the inequality as x2 + x – 6 ≤ 0. Then make a table of values.
Notice that x2 + x –6 ≤ 0 when the values of x are between –3 and 2, inclusive.
The solution of the inequality is –3 ≤ x ≤ 2.
ANSWER

19. EXAMPLE 5
Solve a quadratic inequality by graphing
Solve 2x2 + x – 4 ≥ 0 by graphing.
SOLUTION
The solution consists of the x-values for which the graph of y = 2x2 + x – 4 lies on or above the x-axis. Find the graph’s x-intercepts by letting y = 0 and using the quadratic formula to solve for x.
0 = 2x2 + x – 4
x =
– – 4(2)(– 4)
2(2)
x = – 4
x or x –1.69

20. EXAMPLE 5
Solve a quadratic inequality by graphing
Sketch a parabola that opens up and has 1.19 and –1.69 as x-intercepts. The graph lies on or above the x-axis to the left of (and including) x = – 1.69 and to the right of (and including) x = 1.19.
ANSWER
The solution of the inequality is approximately x ≤ – 1.69 or x ≥ 1.19.

21. GUIDED PRACTICE
for Examples 4 and 5
Solve the inequality 2x2 + 2x ≤ 3 using a table and using a graph.
SOLUTION
Rewrite the inequality as 2x2 + 2x – 3 ≤ 0. Then make a table of values. x -3 -2
-1.8
-1.5 -1
0.5
0.8
0.9
22 + 2x – 3 9 1
-0.1
0.42
The solution of the inequality is –1.8 ≤ x ≤
ANSWER

22. EXAMPLE 6
Use a quadratic inequality as a model
Robotics
The number T of teams that have participated in a robot-building competition for high school students can be modeled by
T(x) = 7.51x2 –16.4x , 0 ≤ x ≤ 9
Where x is the number of years since For what years was the number of teams greater than 100?

23. EXAMPLE 6
Use a quadratic inequality as a model
SOLUTION
You want to find the values of x for which:
T(x) > 100
7.51x2 – 16.4x > 100
7.51x2 – 16.4x – 65 > 0
Graph y = 7.51x2 – 16.4x – 65 on the domain 0 ≤ x ≤ 9. The graph’s x-intercept is about 4.2. The graph lies above the x-axis when 4.2 < x ≤ 9.
There were more than 100 teams participating in the years 1997–2001.
ANSWER

24. Write equation that corresponds to original inequality.
EXAMPLE 7
Solve a quadratic inequality algebraically
Solve x2 – 2x > 15 algebraically.
SOLUTION
First, write and solve the equation obtained by replacing > with = .
x2 – 2x = 15
Write equation that corresponds to original inequality.
x2 – 2x – 15 = 0
Write in standard form.
(x + 3)(x – 5) = 0
Factor.
x = – 3 or x = 5
Zero product property

25. EXAMPLE 7
Solve a quadratic inequality algebraically
The numbers – 3 and 5 are the critical x-values of the inequality x2 – 2x > 15. Plot – 3 and 5 on a number line, using open dots because the values do not satisfy the inequality. The critical x-values partition the number line into three intervals. Test an x-value in each interval to see if it satisfies the inequality.
Test x = – 4:
Test x = 1:
Test x = 6:
(– 4)2 –2(– 4) = 24 >15 
12 –2(1) 5 –1 >15
62 –2(6) = 24 >15 
The solution is x < – 3 or x > 5.
ANSWER

26. GUIDED PRACTICE
for Examples 6 and 7
Robotics
Use the information in Example 6 to determine in what years at least 200 teams participated in the robot-building competition.
SOLUTION
You want to find the values of x for which:
T(x) > 200
7.51x2 – 16.4x > 200
7.51x2 – 16.4x – 165 > 0

27. GUIDED PRACTICE
for Examples 6 and 7
Graph y = 7.51x2 – 16.4x – 165 on the domain 0 ≤ x ≤ 9.
There were more than 200 teams participating in the years 1998 – 2001.
ANSWER

28. Write equation that corresponds to original inequality.
GUIDED PRACTICE
for Examples 6 and 7
Solve the inequality 2x2 – 7x = 4 algebraically.
SOLUTION
First, write and solve the equation obtained by replacing > with 5.
2x2 – 7x = 4
Write equation that corresponds to original inequality.
2x2 – 7x – 4 = 0
Write in standard form.
(2x + 1)(x – 4) = 0
Factor.
x = – 0.5 or x = 4
Zero product property

29. – 3 – 4 – 2 – 1 1 2 3 4 5 6 7 – 5 – 6 – 7 GUIDED PRACTICE
for Examples 6 and 7
The numbers 4 and – 0.5 are the critical x-values of the inequality 2x2 – 7x > 4 . Plot 4 and – 0.5 on a number line, using open dots because the values do not satisfy the inequality. The critical x-values partition the number line into three intervals. Test an x-value in each interval to see if it satisfies the inequality.
– 3
– 4
– 2
– 1 1 2 3 4 5 6 7
– 5
– 6
– 7
Test x = – 3:
Test x = 2:
Test x = 5:
2 (– 3)2 – 7 (– 3) > 4 
2 (2)2 – 7 (2) > 4
2 (5)2 – 7 (3) > 4 
The solution is x < – 0.5 or x > 4.
ANSWER