5-6
Quadratic Equations
TEKS FOCUS
VOCABULARY
ĚZero of a function – A zero of a function f(x) is
TEKS (4)(F) Solve quadratic and square root equations.
TEKS (1)(C) Select tools, including real objects, manipulatives,
paper and pencil, and technology as appropriate, and techniques,
including mental math, estimation, and number sense as
appropriate, to solve problems.
Additional TEKS (1)(A), (1)(E), (7)(I)
any value for which f(x) = 0.
ĚZero-Product Property – If the product of two or
more factors is zero, then one of the factors must
be zero.
ĚNumber sense – the understanding of what
numbers mean and how they are related
ESSENTIAL UNDERSTANDING
To find the zeros of a quadratic function y = ax2 + bx + c, solve the related quadratic
equation 0 = ax2 + bx + c.
Property
Zero-Product Property
If ab = 0, then a = 0 or b = 0.
Problem 1
P
TEKS Process Standard (1)(E)
Solving a Quadratic Equation by Factoring
What do you know
about the factors of
x2 − bx + c?
The product of their
constant terms is c. The
sum is -b.
184
Lesson 5-6
What are the solutions of the quadratic equation x2 − 5x + 6 = 0?
W
(x - 2)(x - 3) = 0
Factor the quadratic expression.
x-2=0
or
x-3=0
Use the Zero-Product Property.
x=2
or
x=3
The
T solutions are x = 2 and x = 3.
Quadratic Equations
Solve for x.
Problem 2
P
TEKS Process Standard (1)(C)
Solving a Quadratic Equation With Tables
What are the solutions of the quadratic equation 5x2 + 30x + 14 = 2 − 2x?
5x2 + 30x + 14 = 2 - 2x
5x2 + 32x + 12 = 0
Rewrite in standard form.
Use your calculator’s TABLE feature to find the zeros.
What should you look
for in the calculator
table?
Look for x-values for
which y = 0.
Plot1
Plot2
Plot3
\Y1 = 55X
X 2+32X+12
\Y2 =
\Y3 =
\Y4 = Enter the
\Y5 = equation in
\Y6 =
\Y7 = standard form
as Y1.
X
–6
–5
–4
–3
–2
–1
0
X = -6
Y1
0
–23
23
–36
–39
–32
–15
12
Y1 = 0, x = –6
is one zero.
x-interval
changed to .1.
X
–.8
–.7
–.6
–.5
–.4
–.3
–.2
X = -.4
Y1
–10.4
–7.95
7.95
–5.4
–2.75
–2.7
75
0
2.85
85
5.8
Y1 = 0, x = –.4
is the second zero.
Second zero is between
x = –1 and x = 0. Notice
change in sign for y-values.
The solutions are x = -6 and x = -0.4.
Problem
bl
3
Solving a Quadratic Equation by Graphing
What are the solutions of the quadratic equation 2x2 + 7x = 15?
2x2 + 7x = 15
2x2 + 7x - 15 = 0
How can you use a
graph to find the
solutions?
Find the zeros of the
related quadratic
function.
Plot1
Plot2
Plot3
\Y1 = 2X 2+7X–15
\Y2 =
\Y3 =
Enter the
\Y4 =
\Y5 =
equation in
\Y6 =
standard form
\Y7 =
as Y1.
Rewrite in standard form.
Use ZERO option
in CALC feature.
Zero
X=–55
Y=0
Zer
Zero
eroo
X=1.5
Y=0
The solutions are x = -5 and x = 1.5.
PearsonTEXAS.com
185
Problem 4
P
Using a Quadratic Equation
Competition From the
time Mark Twain wrote The
Celebrated Jumping Frog of
Calaveras County in 1865,
frog-jumping competitions
have been growing in
popularity. The graph shows a
function modeling the height
of one frog’s jump, where x is
the distance, in feet, from the
jump’s start.
A How far did the frog jump?
How can you use a
graphing calculator
to determine the
distance?
Graph the function and
locate the point where
the graph crosses the
x-axis.
The height of the jump is 0 at the start and
end of the jump. Find the zeros of the function.
Use a graphing calculator to find the zeros of the
related function y = -0.029x2 + 0.59x.
In the air
Plot1
Plot2
\Y1 = –.029X 2+.59X
\Y2 =
\Y3 =
\Y4 =
\Y5 =
\Y6 =
Plot3
Take-off
Land
Nothing past this point
has real-world meaning.
x = 0 at take-off
Zero
X=20.34
x = 20.34
at landing
Y=0
The frog jumped about 20.34 ft.
B How high did the frog jump?
The maximum height of the jump is the maximum value of the function. This occurs
midway, at 10.17 ft from the start. Find y for x = 10.17.
y = -0.029(10.17)2 + 0.59(10.17) ≈ 3.0
The frog jumped to a height of about 3.0 ft.
C What is a reasonable domain and range for such a frog-jumping function?
While the function y = -0.029x2 + 0.59x has a domain of all real numbers,
actual frog jumping does not allow negative values. So, a reasonable domain for
frog-jumping distances is 0 … x … 30. A reasonable range is 0 … y … 5.
186
Lesson 5-6
Quadratic Equations
HO
ME
RK
O
NLINE
WO
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
Solve each equation by factoring. Check your answers.
For additional support when
completing your homework,
go to PearsonTEXAS.com.
1. x2 + 6x + 8 = 0
2. x2 + 18 = 9x
3. 2x2 - x = 3
4. x2 - 10x + 25 = 0
5. 6x2 + 4x = 0
6. 2x2 = 8x
Solve each equation using tables. Give each answer to at most two decimal places.
7. x2 + 5x + 3 = 0
8. x2 - 7x = 11
9. 2x2 - x = 2
Solve each equation by graphing. Give each answer to at most two decimal places.
10. 6x2 = -19x - 15
11. 3x2 - 5x - 4 = 0
12. 5x2 - 7x - 3 = 8
13. Evaluate Reasonableness (1)(B) A classmate solves the quadratic
equation as shown. Find and correct the error. What are the correct
solutions?
14. Create Representations to Communicate Mathematical Ideas (1)(E)
Write an equation with the given solutions.
a. 3 and 5
b. -3 and 2
x2 + 5x + 6 = 2
(x + 2) (x + 3) = 2
x = -2 or x = -3
c. -1 and -6
15. Use Representations to Communicate Mathematical Ideas (1)(E) The function
h = -16t 2 + 1700 gives an object’s height h, in feet, at t seconds.
a. What does the constant 1700 tell you about the height of the object?
b. What does the coefficient of t 2 tell you about the direction the object is moving?
c. When will the object be 1000 ft above the ground?
d. When will the object be 940 ft above the ground?
e. What are a reasonable domain and range for the function h?
16. Apply Mathematics (1)(A) Suppose you want to put a frame
around the painting shown at the right. The frame will be the
same width around the entire painting. You have 276 in.2 of
framing material. How wide should the frame be?
17. The period of a pendulum is the time the pendulum takes
to swing back and forth. The function L = 0.81t 2 relates the
length L in feet of a pendulum to the time t in seconds that
it takes to swing back and forth. A convention center has a
pendulum that is 90 feet long. Find the period.
16 in.
24 in.
18. Apply Mathematics (1)(A) Suppose you have an outdoor pool measuring 25 ft by
10 ft. You want to add a cement walkway around the pool. If the walkway will be 1 ft
thick and you have 304 ft3 of cement, how wide should the walkway be?
PearsonTEXAS.com
187
Select Tools to Solve Problems (1)(C) Solve each equation by factoring, using
tables, or by graphing. If necessary, round your answer to the nearest hundredth.
19. x2 + 2x = 6 - 6x
20. 6x2 + 13x + 6 = 0
21. 2x2 + x - 28 = 0
22. (x + 3)2 = 9
23. x2 + 4x = 0
24. x2 = 8x - 7
25. x2 - 3x = 6
26. 4x2 + 5x = 4
27. 7x - 3x2 = -10
Connect Mathematical Ideas (1)(F) The graphs of each pair of functions intersect.
Find their points of intersection without using a calculator. (Hint: Solve as a
system using substitution.)
28. y = x2
29. y = x2 - 2
3
y = - 12x2 + 2x + 3
30. y = -x2 + x + 4
y = 3x2 - 4x - 2
y = 2x2 - 6
31. The equation x2 - 10x + 24 = 0 can be written in factored form as (x - 4)(x - 6) = 0.
How can you use this fact to find the vertex of the graph of y = x2 - 10x + 24?
32. a. Let a 7 0. Use algebraic or arithmetic ideas to explain why the lowest point on
the graph of y = a(x - h)2 + k must occur when x = h.
b. Suppose that the function in part (a) is y = a(x - h)3 + k. Is your reasoning still
valid? Explain.
33. Apply Mathematics (1)(A) When serving in tennis, a player tosses the tennis ball
vertically in the air. The height h of the ball after t seconds is given by the quadratic
function h(t) = -5t 2 + 7t (the height is measured in meters from the point of the toss).
a. How high in the air does the ball go?
b. Assume that the player hits the ball on its way down when it’s 0.6 m above the point
of the toss. For how many seconds is the ball in the air between the toss and the serve?
TEXAS Test Practice
T
34. What are the solutions of the equation 6x 2 + 9x - 15 = 0?
A. 1, -15
5
F. ( -1, 7)
G. (3, 7)
5
B. 1, - 2
C. -1, -5
D. 3, 2
35. The vertex of a parabola is (3, 2). A second point on the parabola is (1, 7). Which
point is also on the parabola?
H. (5, 7)
J. (3, -2)
36. For which quadratic function is -3 the constant term?
A. y = (3x + 1)( -x - 3)
C. f (x) = (x - 3)(x - 3)
B. y = x 2 - 3x + 3
D. g(x) = -3x 2 + 3x + 9
37. What transformations are needed to go from the parent function f (x) = x2 to the
new function g(x) = -3x2 + 2? Graph g(x).
188
Lesson 5-6
Quadratic Equations
Activity Lab
USE WITH LESSON 5-6
Writing Equations From Roots
TEKS (4)(B), (1)(F)
A root of an equation is a value that makes the equation true. You can use the
Zero-Product Property to write a quadratic function from its zeros or a quadratic
equation from its roots.
1
1.
1 a.
b.
2. a.
b.
c.
Write
W i a nonzero linear function f (x) that has a zero at x = 3.
Write a nonzero linear function g(x) that has a zero at x = 4.
For f and g from Exercise 1, write the product function h(x) = f (x)
What kind of function is h(x)?
Solve the equation h(x) = 0.
# g(x).
Mental Math Write a quadratic equation with each pair of values as roots.
3. 5 and 3
4. 2.5 and 4
6. 5 and 10
3
7. 2 and -2
5. -4 and 4
You can also use zeros or roots to write quadratic expressions in standard form.
2
a
8.
and complete the table. Write the product
8 a. Copy
C
(x - a)(x - b) in standard form for each pair a and b.
b. Is there a pattern in the table? Explain.
9. a. If you know the roots, you can write a quadratic function or
equation in standard form. Explain how.
b. Demonstrate your method for each pair of values in
Exercises 3–7.
b
aⴙb
ab
(x ⴚ a) (x ⴚ b)
x2 ⫺ 9x ⫹ 20
4
5
9
20
⫺4
5
1
⫺20
■
4
⫺5
■
■
■
⫺4
⫺5
■
■
■
⫺9
⫺1
■
■
■
⫺2
7
■
■
■
Exercises
10. Explain how to write a quadratic equation that has -6 as its only root.
11. Describe the family of quadratic functions that have zeros at r and s. Sketch
several members of the family in the coordinate plane.
Find the sum and product of the roots for each quadratic equation.
12. 2x 2 + 3x - 2 = 0
13. x 2 - 2x + 1 = 0
14. x 2 - 5x + 6 = 0
Given the sum and product of the roots, write a quadratic equation in
standard form.
15. sum = -3, product = -18
16. sum = 4, product = 3
3
17. sum = 2, product = 4
PearsonTEXAS.com
189