Download Section 2-5 Quadratic Equations and Inequalities

2-5 Quadratic Equations and Inequalities
61. (a ⫹ bi)(c ⫹ di)
62.
a ⫹ bi
c ⫹ di
63. Show that i4k ⫽ 1, k a natural number.
64. Show that i4k⫹1 ⫽ i, k a natural number.
65. Show that 2 ⫺ i and ⫺2 ⫹ i are square roots of 3 ⫺ 4i.
66. Show that ⫺3 ⫹ 2i and 3 ⫺ 2i are square roots of 5 ⫺ 12i.
67. Describe how you could find the square roots of 8 ⫺ 6i
without using a graphing utility. What are the square roots
of 8 ⫺ 6i?
68. Describe how you could find the square roots of 2i without
using a graphing utility. What are the square roots of 2i?
69. Let Sn ⫽ i ⫹ i2 ⫹ i3 ⫹ ⭈ ⭈ ⭈ ⫹ in, n ⱖ 1. Describe the possible values of Sn.
70. Let Tn ⫽ i2 ⫹ i4 ⫹ i6 ⫹ ⭈ ⭈ ⭈ ⫹ i2n, n ⱖ 1. Describe the
possible values of Tn.
Supply the reasons in the proofs for the theorems stated in
Problems 71 and 72.
71. Theorem: The complex numbers are commutative under
addition.
Proof:
Let a ⫹ bi and c ⫹ di be two arbitrary complex
numbers; then,
Statement
1. (a ⫹ bi) ⫹ (c ⫹ di) ⫽ (a ⫹ c) ⫹ (b ⫹ d)i
2.
⫽ (c ⫹ a) ⫹ (d ⫹ b)i
3.
⫽ (c ⫹ di) ⫹ (a ⫹ bi)
145
Reason
1.
2.
3.
72. Theorem: The complex numbers are commutative under
multiplication.
Proof:
Let a ⫹ bi and c ⫹ di be two arbitrary complex
numbers; then,
Statement
1. (a ⫹ bi) ⴢ (c ⫹ di) ⫽ (ac ⫺ bd) ⫹ (ad ⫹ bc)i
2.
⫽ (ca ⫺ db) ⫹ (da ⫹ cb)i
3.
⫽ (c ⫹ di)(a ⫹ bi)
Reason
1.
2.
3.
Letters z and w are often used as complex variables, where
z ⫽ x ⫹ yi, w ⫽ u ⫹ vi, and x, y, u, and v are real numbers.
The conjugates of z and w, denoted by z and w, respectively,
are given by z ⫽ x ⫺ yi and w ⫽ u ⫺ vi. In Problems 73–80,
express each property of conjugates verbally and then prove
the property.
73. zz is a real number.
74. z ⫹ z is a real number.
75. z ⫽ z if and only if z is real. 76. z ⫽ z
77. z ⫹ w ⫽ z ⫹ w
78. z ⫺ w ⫽ z ⫺ w
79. zw ⫽ z ⴢ w
80. z/w ⫽ z/ w
Section 2-5 Quadratic Equations and Inequalities
Introduction
Solution by Factoring
Solution by Completing the Square
Solution by Quadratic Formula
Solving Quadratic Inequalities
Introduction
In this book we are primarily interested in functions with real number domains
and ranges. However, if we want to fully understand the nature of the zeros of a
function or the roots of an equation, it is necessary to extend some of the definitions in Section 1-4 to include complex numbers. A complex number r is a zero
146
2 LINEAR AND QUADRATIC FUNCTIONS
of the function f(x) and a root of the equation f(x) ⫽ 0 if f(r) ⫽ 0. As before, if
r is a real number, then r is also an x intercept of the graph of f. An imaginary
zero can never be an x intercept.
If a, b, and c are real numbers, a ⫽ 0, then associated with the quadratic
function
f(x) ⫽ ax2 ⫹ bx ⫹ c
is the quadratic equation
ax2 ⫹ bx ⫹ c ⫽ 0
Explore/Discuss
1
Match the zeros of each function with one of the sets A, B, or C:
Function
Zeros
f(x) ⫽ x2 ⫺ 1
A ⫽ {1}
g(x) ⫽ x ⫹ 1
B ⫽ {⫺1, 1}
h(x) ⫽ (x ⫺ 1)2
C ⫽ {⫺i, i}
2
Which of these sets of zeros can be found using graphical approximation
techniques? Which cannot?
A graphing utility can be used to approximate the real roots of an equation,
but not the imaginary roots. In this section we will develop algebraic techniques
for finding the exact value of the roots of a quadratic equation, real or imaginary.
In the process, we will derive the well-known quadratic formula, another important tool for our mathematical toolbox.
Solution by Factoring
If ax2 ⫹ bx ⫹ c can be written as the product of two first-degree factors, then
the quadratic equation can be quickly and easily solved. The method of solution
by factoring rests on the zero property of complex numbers, which is a generalization of the zero property of real numbers introduced in Section A-1.
ZERO PROPERTY
If m and n are complex numbers, then
mⴢn⫽0
EXAMPLE
1
if and only if
m ⫽ 0 or n ⫽ 0 (or both)
Solving Quadratic Equations by Factoring
Solve by factoring:
(A) 6x2 ⫺ 19x ⫺ 7 ⫽ 0
(B) x2 ⫺ 6x ⫹ 5 ⫽ ⫺4
(C) 2x2 ⫽ 3x
2-5 Quadratic Equations and Inequalities
Solutions
(A)
147
6x2 ⫺ 19x ⫺ 7 ⫽ 0
(2x ⫺ 7)(3x ⫹ 1) ⫽ 0
2x ⫺ 7 ⫽ 0
Factor left side.
3x ⫹ 1 ⫽ 0
or
x ⫽ 72
x ⫽ ⫺ 13
The solution set is {⫺ 13, 72}.
(B) x2 ⫺ 6x ⫹ 5 ⫽ ⫺4
x2 ⫺ 6x ⫹ 9 ⫽ 0
(x ⫺ 3) ⫽ 0
2
Write in standard form.
Factor left side.
x⫽3
The solution set is {3}. The equation has one root, 3. But since it came from
two factors, we call 3 a double root or a root of multiplicity 2.
(C)
2x2 ⫽ 3x
2x2 ⫺ 3x ⫽ 0
x(2x ⫺ 3) ⫽ 0
x⫽0
or
2x ⫺ 3 ⫽ 0
x⫽
3
2
Solution set: {0, 32}
MATCHED PROBLEM
1
CAUTION
Solve by factoring:
(A) 3x2 ⫹ 7x ⫺ 20 ⫽ 0
(B) 4x2 ⫹ 12x ⫹ 9 ⫽ 0
(C) 4x2 ⫽ 5x
1. One side of an equation must be 0 before the zero property can be
applied. Thus
x2 ⫺ 6x ⫹ 5 ⫽ ⫺4
(x ⫺ 1)(x ⫺ 5) ⫽ ⫺4
does not imply that x ⫺ 1 ⫽ ⫺4 or x ⫺ 5 ⫽ ⫺4. See Example 1,
part B, for the correct solution of this equation.
2. The equations
2x2 ⫽ 3x
and
2x ⫽ 3
are not equivalent. The first has solution set {0, 32}, while the second
has solution set {32}. The root x ⫽ 0 is lost when each member of
the first equation is divided by the variable x. See Example 1, part C,
for the correct solution of this equation.
148
2 LINEAR AND QUADRATIC FUNCTIONS
Do not divide both members of an equation by an expression containing the variable for which you are solving. You may be dividing by 0.
Remark
It is common practice to represent solutions of quadratic equations informally by
the last equation rather than by writing a solution set using set notation. From
now on, we will follow this practice unless a particular emphasis is desired.
Solution by Completing the Square
Factoring is a specialized method that is very efficient if the factors can be quickly
identified. However, not all quadratic equations are easy to factor. We now turn
to a more general process that is guaranteed to work in all cases. This process is
based on completing the square, discussed in Section 2-3, and the following
square root property:
SQUARE ROOT PROPERTY
For any complex numbers r and s, if r2 ⫽ s, then r ⫽ ⫾兹s.
EXAMPLE
2
Solutions
Solution by Completing the Square
Use completing the square and the square root property to solve each of the
following:
(C) 2x2 ⫺ 4x ⫹ 3 ⫽ 0
(A) (x ⫹ 12)2 ⫺ 54 ⫽ 0
(B) x2 ⫹ 6x ⫺ 2 ⫽ 0
(A) This quadratic expression is already written in standard form. We solve for
the squared term and then use the square root property:
(x ⫹ 12)2 ⫺ 54 ⫽ 0
(x ⫹ 12)2 ⫽ 54
Apply the square root property.
x ⫹ ⫽ ⫾兹
1
2
5
4
Solve for x.
1 兹5
x⫽⫺ ⫾
2
2
⫽
⫺1 ⫾ 兹5
2
(B) We can speed up the process of completing the square by taking advantage
of the fact that we are working with a quadratic equation, not a quadratic
expression.
x2 ⫹ 6x ⫺ 2 ⫽ 0
x2 ⫹ 6x ⫽ 2
x2 ⫹ 6x ⴙ 9 ⫽ 2 ⴙ 9
(x ⫹ 3) ⫽ 11
2
x ⫹ 3 ⫽ ⫾兹11
x ⫽ ⫺3 ⫾ 兹11
Complete the square on the
left side, and add the same
number to the right side.
2-5 Quadratic Equations and Inequalities
(C)
149
2x2 ⫺ 4x ⫹ 3 ⫽ 0
x2 ⫺ 2x ⫹ 32 ⫽ 0
Make the leading coefficient 1 by
dividing by 2.
x2 ⫺ 2x ⫽ ⫺ 32
x2 ⫺ 2x ⴙ 1 ⫽ ⫺ 32 ⴙ 1
(x ⫺ 1)2 ⫽ ⫺ 12
Complete the square on the left side and
add the same number to the right side.
Factor the left side.
x ⫺ 1 ⫽ ⫾兹⫺ 12
x ⫽ 1 ⫾ i兹12
⫽1⫾
MATCHED PROBLEM
2
Explore/Discuss
2
兹2
i
2
Answer in a ⫹ bi form.
Solve by completing the square:
(A) (x ⫹ 13)2 ⫺ 29 ⫽ 0
(B) x2 ⫹ 8x ⫺ 3 ⫽ 0
(C) 3x2 ⫺ 12x ⫹ 13 ⫽ 0
Graph the quadratic functions associated with the three quadratic
equations in Example 2. Approximate the x intercepts of each function
and compare with the roots found in Example 2. Which of these
equations has roots that cannot be approximated graphically?
Solution by Quadratic Formula
Now consider the general quadratic equation with unspecified coefficients:
ax2 ⴙ bx ⴙ c ⴝ 0
aⴝ0
We can solve it by completing the square exactly as we did in Example 2, part
C. To make the leading coefficient 1, we must multiply both sides of the equation by 1/a. Thus,
b
c
x2 ⫹ x ⫹ ⫽ 0
a
a
Adding ⫺c/a to both sides of the equation and then completing the square of the
left side, we have
b
b2
b2
c
x2 ⫹ x ⫹ 2 ⫽ 2 ⫺
a
4a
4a
a
150
2 LINEAR AND QUADRATIC FUNCTIONS
We now factor the left side and solve using the square root property:
冢
x⫹
b
2a
x⫹
冣
2
⫽
b2 ⫺ 4ac
4a2
b
⫽⫾
2a
x⫽⫺
⫽
兹
b2 ⫺ 4ac
4a2
b
兹b2 ⫺ 4ac
⫾
2a
2a
See Problem 77.
⫺b ⫾ 兹b2 ⫺ 4ac
2a
We have thus derived the well-known and widely used quadratic formula:
THEOREM
1
QUADRATIC FORMULA
If ax2 ⫹ bx ⫹ c ⫽ 0, a ⫽ 0, then
xⴝ
ⴚb ⴞ 兹b2 ⴚ 4ac
2a
The quadratic formula and completing the square are equivalent methods.
Either can be used to find the exact value of the roots of any quadratic equation.
EXAMPLE
3
Using the Quadratic Formula
Solve 2x ⫹ 32 ⫽ x2 by use of the quadratic formula. Leave the answer in simplest radical form.
2x ⫹ 32 ⫽ x2
Solution
4x ⫹ 3 ⫽ 2x2
2x2 ⫺ 4x ⫺ 3 ⫽ 0
⫺b ⫾ 兹b ⫺ 4ac
2a
Multiply both sides by 2.
Write in standard form.
2
x⫽
CAUTION
a ⫽ 2, b ⫽ ⫺4, c ⫽ ⫺3
⫽
⫺(⫺4) ⫾ 兹(⫺4)2 ⫺ 4(2)(⫺3)
2(2)
⫽
4 ⫾ 兹40 4 ⫾ 2兹10 2 ⫾ 兹10
⫽
⫽
4
4
2
1. ⫺42 ⫽ (⫺4)2 ⫺42 ⫽ ⫺16 and (⫺4)2 ⫽ 16
兹10 2 ⫹ 兹10
兹10 4 ⫹ 兹10
2. 2 ⫹
⫽
2⫹
⫽
2
2
2
2
4 ⫾ 2兹10
4 ⫾ 2兹10 2(2 ⫾ 兹10) 2 ⫾ 兹10
3.
⫽ ⫾2兹10
⫽
⫽
4
4
2
4
2-5 Quadratic Equations and Inequalities
MATCHED PROBLEM
3
151
Solve x2 ⫺ 52 ⫽ ⫺3x using the quadratic formula. Leave the answer in simplest
radical form.
Given the quadratic function f(x) ⫽ ax2 ⫹ bx ⫹ c, let D ⫽ b2 ⫺ 4ac.
How many real zeros does f have if
(A) D ⬎ 0
(B) D ⫽ 0
(C) D ⬍ 0
In each of these three cases, what type of roots does the quadratic equation f(x) ⫽ 0 have?
Explore/Discuss
3
The quantity b2 ⴚ 4ac in the quadratic formula is called the discriminant and
gives us information about the roots of the corresponding equation and the zeros
of the associated quadratic function. This information is summarized in Table 1.
T A B L E
1 Discriminants, Roots, and Zeros
Roots of*
ax2 ⴙ bx ⴙ c ⴝ 0
Discriminant
b ⴚ 4ac
2
Number of Real Zeros of*
f (x) ⴝ ax2 ⴙ bx ⴙ c
Two distinct real roots
One real root (a double root)
Two imaginary roots, one the
conjugate of the other
Positive
0
Negative
2
1
0
*a, b, and c are real numbers with a ⫽ 0.
EXAMPLE
Design
4
Solution
A picture frame of uniform width has outer dimensions of 12 inches by 18
inches. How wide (to the nearest tenth of an inch) must the frame be to display an area of 140 square inches?
We begin by drawing and labeling a figure:
x
12
12 ⫺ 2x
18 ⫺ 2x
18
If x is the width of the frame, then x must satisfy the equation
(18 ⫺ 2x)(12 ⫺ 2x) ⫽ 140
(1)
152
2 LINEAR AND QUADRATIC FUNCTIONS
Note that x must satisfy 0 ⱕ x ⱕ 6 to insure that both 12 ⫺ 2x and 18 ⫺ 2x are
nonnegative. The roots of this quadratic equation can be found algebraically or
approximated graphically. Using both methods will confirm that we have the correct answer. We begin with the algebraic solution:
(18 ⫺ 2x)(12 ⫺ 2x) ⫽ 140
216 ⫺ 36x ⫺ 24x ⫹ 4x2 ⫽ 140
4x2 ⫺ 60x ⫹ 76 ⫽ 0
x2 ⫺ 15x ⫹ 19 ⫽ 0
x⫽
Thus, the quadratic equation has two solutions (rounded to one decimal place):
FIGURE 1
250
0
15 ⫾ 兹149
2
x⫽
15 ⫹ 兹149
⫽ 13.6
2
and
x⫽
15 ⫺ 兹149
⫽ 1.4
2
6
The first must be discarded as being much too large. So the width of the frame
is 1.4 inches.
Graphing both sides of equation (1) for 0 ⱕ x ⱕ 6 and using an intersection
routine confirms that this answer is correct (Fig. 1).
MATCHED PROBLEM
A 1,200 square foot garden is enclosed with 150 feet of fencing. Find the dimensions of the garden to the nearest tenth of a foot.
0
4
Solving Quadratic Inequalities
Explore/Discuss
4
Graph f(x) ⫽ (x ⫹ 2)(x ⫺ 3) and examine the graph to determine the
solutions of the following inequalities:
(A) f(x) ⬎ 0
(B) f(x) ⬍ 0
(C) f(x) ⱖ 0
(D) f(x) ⱕ 0
The simplest method for solving inequalities involving a function is to find the
zeros of the function and then examine the graph to determine where the function is positive and where it is negative. Inequalities involving quadratic functions
are handled routinely by this method, as the following examples illustrate.
EXAMPLE
5
Solution
Finding the Domain of a Function
Find the domain of f(x) ⫽ 兹x2 ⫺ 4x ⫺ 9. Express answer in interval notation using exact values.
The domain of this function is the set of all real numbers x that produce real values for f(x) (Section 1-3).
2-5 Quadratic Equations and Inequalities
153
This is precisely the solution set of the quadratic inequality
x2 ⫺ 4x ⫺ 9 ⱖ 0
(2)
The solution of this inequality consists of all values of x for which the graph of
y ⫽ x2 ⫺ 4x ⫺ 9 is on or above the x axis. Using either completing the square
or the quadratic formula, we find that the x intercepts are
FIGURE 2
y ⫽ x2 ⫺ 4x ⫺ 9.
15
x⫽
⫺10
4 ⫾ 兹52
⫽ 2 ⫾ 兹13
2
10
Examining the graph in Figure 2, we see that the solution of inequality (2) and,
hence, the domain of f, is
⫺15
2 ⫺ 兹13
(⫺⬁, 2 ⫺ 兹13] 傼 [2 ⫹ 兹13, ⬁)
2 ⫹ 兹13
MATCHED PROBLEM
5
Find the domain of g(x) ⫽ 兹2 ⫹ 2x ⫺ x2. Express answer in interval notation
using exact values.
EXAMPLE
Projectile Motion
6
If a projectile is shot straight upward from the ground with an initial velocity
of 160 feet per second, its distance d (in feet) above the ground at the end of
t seconds (neglecting air resistance) is given approximately by
d(t) ⫽ 160t ⫺ 16t2
(A) What is the domain of d?
(B) At what times (to two decimal places) will the projectile be more than
200 feet above the ground?
Express answers in inequality notation.
Solutions
(A) Factoring d(t), we have
d(t) ⫽ 160t ⫺ 16t2 ⫽ 16t(10 ⫺ t)
Thus, d(0) ⫽ 0 and d(10) ⫽ 0. The projectile is released at t ⫽ 0 seconds and
returns to the ground at t ⫽ 10 seconds, so the domain of d is 0 ⱕ t ⱕ 10.
(B) Since we are asked for two-decimal-place accuracy, we can solve this problem graphically. Graph d and the horizontal line y ⫽ 200 and find the intersection points (Fig. 3).
FIGURE 3
500
0
500
10
0
10
0
0
(a)
(b)
154
2 LINEAR AND QUADRATIC FUNCTIONS
From Figure 3 we see that the projectile will be above 200 feet for
1.46 ⬍ t ⬍ 8.54.
MATCHED PROBLEM
6
Refer to the projectile equation in Example 6. At what times (to two decimal
places) during its flight will the projectile be less than 250 feet above the ground?
Express answer in inequality notation.
Answers to Matched Problems
1. (A) {⫺4, 53}
(B) {⫺ 32} (a double root)
(C) {0, 54}
2. (A) x ⫽ (⫺1 ⫾ 兹2)/3
(B) x ⫽ ⫺4 ⫾ 兹19
(C) x ⫽ (6 ⫾ i兹3)/3 or 2 ⫾ (兹3/3)i
3. x ⫽ (⫺3 ⫾ 兹19)/2
4. 23.1 ft by 51.9 ft
5. [1 ⫺ 兹3, 1 ⫹ 兹3]
6. 0 ⱕ t ⬍ 1.94 or 8.06 ⬍ t ⱕ 10
EXERCISE 2-5
31. x2 ⱕ 8x
32. x2 ⫹ 6x ⱖ 0
33. x2 ⫹ 5x ⱕ 0
34. x2 ⱕ 4x
A
B
In Problems 1–6, solve by factoring.
1. 4u2 ⫽ 8u
2. 3A2 ⫽ ⫺12A
3. 9y ⫽ 12y ⫺ 4
4. 16x ⫹ 8x ⫽ ⫺1
5. 11x ⫽ 2x2 ⫹ 12
6. 8 ⫺ 10x ⫽ 3x2
2
2
In Problems 7–18, solve by using the square root property.
7. m ⫺ 12 ⫽ 0
8. y ⫺ 45 ⫽ 0
9. x ⫹ 25 ⫽ 0
10. x ⫹ 16 ⫽ 0
11. 9y2 ⫺ 16 ⫽ 0
12. 4x2 ⫺ 9 ⫽ 0
13. 4x2 ⫹ 25 ⫽ 0
14. 16a2 ⫹ 9 ⫽ 0
15. (n ⫹ 5)2 ⫽ 9
16. (m ⫺ 3)2 ⫽ 25
17. (d ⫺ 3) ⫽ ⫺4
18. (t ⫹ 1) ⫽ ⫺9
2
2
2
2
2
2
In Problems 19–26, solve using the quadratic formula.
In Problems 35–44, find exact answers and check with a
graphing utility, if possible.
35. x2 ⫺ 6x ⫺ 3 ⫽ 0
36. y2 ⫺ 10y ⫺ 3 ⫽ 0
37. 2y2 ⫺ 6y ⫹ 3 ⫽ 0
38. 2d 2 ⫺ 4d ⫹ 1 ⫽ 0
39. 3x2 ⫺ 2x ⫺ 2 ⫽ 0
40. 3x2 ⫹ 5x ⫺ 4 ⫽ 0
41. 12x2 ⫹ 7x ⫽ 10
42. 9x2 ⫹ 9x ⫽ 4
43. x2 ⫽ 3x ⫹ 1
44. x2 ⫹ 2x ⫽ 2
In Problems 45–48, solve for the indicated variable in terms of
the other variables. Use positive square roots only.
45. s ⫽ 12gt2
46. a2 ⫹ b2 ⫽ c2
for t
47. P ⫽ EI ⫺ RI
2
for I
for a
48. A ⫽ P(1 ⫹ r)
2
for r
19. x2 ⫺ 10x ⫺ 3 ⫽ 0
20. x2 ⫺ 6x ⫺ 3 ⫽ 0
In Problems 49–52, solve to two decimal places. Express
answers in inequality notation.
21. x2 ⫹ 8 ⫽ 4x
22. y2 ⫹ 3 ⫽ 2y
49. 2.07x2 ⫺ 3.79x ⫹ 1.34 ⬎ 0
23. 2x2 ⫹ 1 ⫽ 4x
24. 2m2 ⫹ 3 ⫽ 6m
50. 0.61x2 ⫺ 4.28x ⫹ 2.93 ⬍ 0
25. 5x2 ⫹ 2 ⫽ 2x
26. 7x2 ⫹ 6x ⫹ 4 ⫽ 0
51. 4.83x2 ⫹ 2.04x ⫺ 3.18 ⱕ 0
In Problems 27–34, solve and graph. Express answers in both
inequality and interval notation.
52. 5.13x2 ⫹ 7.27x ⫺ 4.32 ⱖ 0
27. x2 ⬍ 10 ⫺ 3x
28. x2 ⫹ x ⬍ 12
In Problems 53–60, find the domain of each function. Express
answers in interval notation using exact values.
29. x2 ⫹ 21 ⬎ 10x
30. x2 ⫹ 7x ⫹ 10 ⬎ 0
53. f(x) ⫽ 兹x2 ⫺ 9
54. g(x) ⫽ 兹4 ⫺ x2
2-5 Quadratic Equations and Inequalities
4
55. h(x) ⫽ 兹
2x2 ⫹ x ⫺ 6
57. F(x) ⫽
1
兹6x ⫺ x2 ⫺ 4
6
56. k(x) ⫽ 兹
3x2 ⫺ 7x ⫺ 6
58. G(x) ⫽
1
兹8x ⫺ x2 ⫺ 14
59. Consider the quadratic equation
x2 ⫹ 4x ⫹ c ⫽ 0
where c is a real number. Discuss the relationship between
the values of c and the three types of roots listed in Table 1.
60. Consider the quadratic equation
x2 ⫺ 2x ⫹ c ⫽ 0
where c is a real number. Discuss the relationship between
the values of c and the three types of roots listed in Table 1.
In Problems 61–64, use the given information concerning the
roots of the quadratic equation ax2 ⫹ bx ⫹ c ⫽ 0, a ⫽ 0, to
describe the possible solution sets for the indicated inequality.
Illustrate your conclusions with specific examples.
61. ax2 ⫹ bx ⫹ c ⬎ 0, given distinct real roots r1 and r2 with
r1 ⬍ r2.
62. ax2 ⫹ bx ⫹ c ⱕ 0, given distinct real roots r1 and r2 with
r1 ⬍ r2.
63. ax2 ⫹ bx ⫹ c ⱖ 0, given one (double) real root r.
64. ax2 ⫹ bx ⫹ c ⬍ 0, given one (double) real root r.
65. Give an example of a quadratic inequality whose solution
set is the entire real line.
66. Give an example of a quadratic inequality whose solution
set is the empty set.
155
⫾兹(b2 ⫺ 4ac)/4a2
with
⫾兹b2 ⫺ 4ac/2a
What justifies using 2a in place of 兩2a兩?
78. Find the error in the following “proof” that two arbitrary
numbers are equal to each other: Let a and b be arbitrary
numbers such that a ⫽ b. Then
(a ⫺ b)2 ⫽ a2 ⫺ 2ab ⫹ b2 ⫽ b2 ⫺ 2ab ⫹ a2
(a ⫺ b)2 ⫽ (b ⫺ a)2
a⫺b⫽b⫺a
2a ⫽ 2b
a⫽b
APPLICATIONS
79. Numbers. Find two numbers such that their sum is 21 and
their product is 104.
80. Numbers. Find all numbers with the property that when
the number is added to itself the sum is the same as when
the number is multiplied by itself.
81. Numbers. Find two consecutive positive even integers
whose product is 168.
82. Numbers. Find two consecutive positive integers whose
product is 600.
83. Profit Analysis. A screen printer produces custom silkscreen apparel. The cost C(x) of printing x custom T-shirts
and the revenue R(x) from the sale of x T-shirts (both in
dollars) are given by
C(x) ⫽ 200 ⫹ 2.25x
C
R(x) ⫽ 10x ⫺ 0.05x2
Solve Problems 67–70 and express answer in a ⫹ bi form.
67. x2 ⫹ 3ix ⫺ 2 ⫽ 0
68. x2 ⫺ 7ix ⫺ 10 ⫽ 0
69. x2 ⫹ 2ix ⫽ 3
70. x2 ⫽ 2ix ⫺ 3
Determine the production levels x (to the nearest integer)
that will result in the printer showing a profit.
In Problems 71 and 72, find all solutions.
71. x3 ⫺ 1 ⫽ 0
72. x4 ⫺ 1 ⫽ 0
73. Can a quadratic equation with rational coefficients have
one rational root and one irrational root? Explain.
74. Can a quadratic equation with real coefficients have one
real root and one imaginary root? Explain.
75. Show that if r1 and r2 are the two roots of ax2 ⫹ bx ⫹ c ⫽
0, then r1r2 ⫽ c/a.
76. For r1 and r2 in Problem 75, show that r1 ⫹ r2 ⫽ ⫺b/a.
84. Profit Analysis. Refer to Problem 83. Determine the production levels x (to the nearest integer) that will result in
the printer showing a profit of at least $60.
77. In one stage of the derivation of the quadratic formula, we
replaced the expression
85. Air Search. A search plane takes off from an airport at 6:00
A.M. and travels due north at 200 miles per hour. A second
156
2 LINEAR AND QUADRATIC FUNCTIONS
plane takes off at 6:30 A.M. and travels due east at 170
miles per hour. The planes carry radios with a maximum
range of 500 miles. When (to the nearest minute) will these
planes no longer be able to communicate with each other?
(B) Building codes require that this building have a crosssectional area of at least 15,000 square feet. What are
the widths of the buildings that will satisfy the
building codes?
(C) Can the developer construct a building with a crosssectional area of 25,000 square feet? What is the
maximum cross-sectional area of a building
constructed in this manner?
86. Projectile Flight. If a projectile is shot straight upward from
the ground with an initial velocity of 176 feet per second, its
distance d (in feet) above the ground at the end of t seconds
(neglecting air resistance) is given approximately by
d(t) ⫽ 176t ⫺ 16t2
★
(A) What is the domain of d?
(B) At what times (to two decimal places) will the
projectile be more than 200 feet above the ground?
90. Architecture. An architect is designing a small A-frame
cottage for a resort area. A cross-section of the cottage is
an isosceles triangle with a base of 5 meters and an altitude
of 4 meters. The front wall of the cottage must accommodate a sliding door positioned as shown in the figure.
Express answers in inequality notation.
87. Construction. A gardener has a 30 foot by 20 foot rectangular plot of ground. She wants to build a brick walkway
of uniform width on the border of the plot (see the figure).
If the gardener wants to have 400 square feet of ground
left for planting, how wide (to two decimal places) should
she build the walkway?
DOOR DETAIL
Page 1 of 4
w
4 meters
h
x
5 meters
20 feet
30 feet
88. Construction. Refer to Problem 87. The gardener buys
enough brick to build 160 square feet of walkway. Is this
sufficient to build the walkway determined in Problem 87?
If not, how wide (to two decimal places) can she build the
walkway with these bricks?
REBEKAH DRIVE
89. Architecture. A developer wants to erect a rectangular
building on a triangular-shaped piece of property that is
200 feet wide and 400 feet long (see the figure).
200 feet
★
Property
A
Property Line
l
Proposed
Building
w
FIRST STREET
(A) Express the area A(w) of the door as a function of the
width w and state the domain of this function. [See
the hint for Problem 89.]
(B) A provision of the building code requires that
doorways must have an area of at least 4.2 square
meters. Find the width of the doorways that satisfy
this provision.
(C) A second provision of the building code requires all
doorways to be at least 2 meters high. Discuss the
effect of this requirement on the answer to part B.
91. Transportation. A delivery truck leaves a warehouse and
travels north to factory A. From factory A the truck travels
east to factory B and then returns directly to the warehouse
(see the figure). The driver recorded the truck’s odometer
reading at the warehouse at both the beginning and the end
of the trip and also at factory B, but forgot to record it at
factory A (see the table). The driver does recall that it was
further from the warehouse to factory A than it was from
factory A to factory B. Since delivery charges are based on
distance from the warehouse, the driver needs to know how
far factory A is from the warehouse. Find this distance.
400 feet
(A) Express the cross-sectional area A(w) of the building
as a function of the width w and state the domain of
this function. [Hint: Use Euclid’s theorem* to find a
relationship between the length l and width w.]
*Euclid’s theorem: If two triangles are similar, their corresponding
sides are proportional:
c
a
b
a⬘
c⬘
b⬘
a
b
c
⫽ ⫽
a⬘ b⬘ c⬘
2-6 Additional Equation Solving Techniques
★★
Factory A
Factory B
157
92. Construction. A 14-mile track for racing stock cars consists
of two semicircles connected by parallel straight-aways
(see the figure). To provide sufficient room for pit crews,
emergency vehicles, and spectator parking, the track must
enclose an area of 100,000 square feet. Find the length of
the straightaways and the diameter of the semicircles to
the nearest foot. [Recall: The area A and circumference C
of a circle of diameter d are given by A ⫽ ␲d 2/4 and
C ⫽ ␲d.]
Warehouse
100,000 square feet
Odometer Readings
Warehouse
5 2 8 4 6
Factory A
5 2 ? ? ?
Factory B
5 2 9 3 7
Warehouse
5 3 0 0 2
Section 2-6 Additional Equation Solving Techniques
Equations Involving Radicals
Equations of the Form ax2p ⫹ bx p ⫹ c ⫽ 0
In this section we examine equations that can be transformed into quadratic equations by various algebraic manipulations. With proper interpretation, the solutions
of the resulting quadratic equations will lead to the solutions of the original
equations.
Equations Involving Radicals
Consider the equation
x ⫽ 兹x ⫹ 2
Graphing both sides of the equation and using an intersection routine shows that
x ⫽ 2 is a solution to the equation (Fig. 1). Is it the only solution?
There may be other solutions not visible in this viewing window. Or there may
be imaginary solutions (remember, graphical approximation applies only to real
solutions). To solve this equation algebraically, we square each side of equation
(1) and then proceed to solve the resulting quadratic equation. Thus,
FIGURE 1
y1 ⫽ x,
y2 ⫽ 兹x ⫹ 2.
5
⫺5
(1)
5
x2 ⫽ (兹x ⫹ 2)2
x ⫽x⫹2
2
⫺5
x2 ⫺ x ⫺ 2 ⫽ 0
(x ⫺ 2)(x ⫹ 1) ⫽ 0
x ⫽ 2, ⫺1
(2)