Objective 8.1A
Vocabulary to Review
quadratic equation [5.7A]
standard form [5.7A]
New Vocabulary
second-degree equation
double root
Rules to Review
Principle of Zero Products
[5.7A]
Discuss the Concepts
1. How does a quadratic equation differ from a linear equation?
2. Explain why the equation x  x  4   12 cannot be solved by solving the two equations x  12 and
x  4  12.
Concept Check
1. How many consecutive natural numbers, beginning with 1, will give a sum of 78? Use the formula
S
n2  n
, where S is the sum of the first n natural numbers.
2
12
2. A football league has 28 games scheduled. How many teams are in the league if each team plays every
other team once? Use the formula N 
t2  t
, where N is the number of football games that must be
2
scheduled in a league having t teams if each team is to play every other team once.
8 teams
Optional Student Activity
1. For how many integer values of b does the equation x2  bx  16  0 have integer solutions?
2. One solution of the equation 2 x2  5x  c  0 is 4. What is the other solution?
2
3
2
3. Given x  t  2t, y  t  t , and x  3, find the possible values of y.

Five
3
2
-18, 2
Objective 8.1B
Concept Check
1. A quadratic equation has two roots, one of which is twice the other. If the sum of the roots is 30, find the
2
quadratic equation and express it in the form ax  bx  c  0, where a, b, and c are integers and c > 0.
x2 – 30x + 200 = 0
2. Marc and Elena tried to solve the same equation, one of the form ax2  bx  c  0. Unfortunately, each
person made just one copying error. The solutions of Marc’sequation were -2 and 
3
but his value of a
2
was incorrect. The solutions of Elena’s equation were 2 and 3 but her value of b was incorrect. Find the
solutions of the correct equation.
-6, -1
Optional Student Activity
Here is an extension of the concepts developed in this objective.
 x  r1  x  r2   x   r  r  x  r r
2
1
2
1 2
Thus the coefficient of x is the opposite of the sum of the roots, and the constant term is the product of the
roots. This result can be used to check the solutions of a quadratic equation.
1. Check that 1  2 and 1  2 are solutions of the equation x2  2 x  1  0.

 

  1  2  1  2   2 , the coefficient of x;

1  2 1  2   1 , the constant term
2. Check that 2  3 and 2  3 are solutions of the equation x2  4 x  1  0.

 

  2  3  2  3   4 , the coefficient of x;


 2  3  2  3   1 , the constant term
Objective 8.1C
New Symbols
a
Discuss the Concepts
2
1. What are the solutions of x  25, and why?
2
2. If A  P(1  r ) then
x2  16 and therefore Explain why the sign is necessary.
3. Given that x400  400400 and x  400 , what is the value of x?
-400
Concept Check
1. The equation d  0.071 y can be used to approximate the distance d, in feet, a car traveling v miles per
hour will skid before stopping after its brakes are applied. An officer investigating an auto accident noted
that the vehicle involved required 144 ft to stop. At what speed was the vehicle traveling before its brakes
were applied? Round to the nearest whole number.
45 mph
2
2. The value A of an initial investment of P dollars after 2 years is given by A  P(1  r ) , where r is the
annual percentage rate earned by the investment. If an initial investment of $5000 grew to a value of
$5724.50 in 2 years, what was the annual percentage rate?
7%
2
Answers to Writing Exercises
1. If a  0 in the equation ax  bx  c  0, then there is no second-degree term in the equation, and it is
therefore not a quadratic equation.
2
2. The Principle of Zero Products states that if the product of two factors is zero, then at least one of the
factors must be zero. This principle is used to solve a quadratic equation of the form ax2  bx  c  0 in
which ax2  bx  c can be written as a product of factors.
Answers to Writing Exercises
127. a.
ax 2  bx  0, a  0, b  0
x(ax  b)  0
x0
ax  b  0
ax  b
x
b
a
The solutions are 0 and 
b
.
a
b.
ax 2  c  0, a  0, c  0
ax 2  c
c
x2  
a
x2  
c
a
x  i
c
a
x  i
c a

a a
x  i
ca
a2
ca
a
ca
x
i
a
x  i
The solutions are
ca
ca
i and 
i.
a
a
Objective 8.2A
Vocabulary to Review
perfect-square trinomial [5.6A]
square of a binomial [5.3C]
New Vocabulary
completing the square
New Equations
(
1
coefficient of x)2  constant term
2
Discuss the Concepts
1. To complete the square on x 2  6 x, what number must be added to the expression? Explain how you
arrived at the answer.
2. If you attempt to solve x2  6x  9  0 by completing the square, the result after simplifying is
 x  3
2
 0. Does this mean that the original equation has no solution? If not, what are the solutions of
the equation?
No. The equation has a double root, 3.
3. What is the next step when using the method of completing the square to solve x2  6x  4?
4. What is the meaning of the phrase “complete the square”?
Concept Check
1. What number is equal to one less than its square?
1 5
1 5
or
2
2
2
2
2. The equation x2  7 x  11  0 has solutions r1 and r2 . Find the value of r1  r2 .
27
Optional Student Activity
2
1. If r1 and r2 are the two solutions of x  5x  3  0, find the value of  r1  r2  r1r2  .
15
2. You have been hired by an observatory to track meteorites and determine whether they will strike Earth.
2
2
The equation of Earth’s path is x  y  40. The first meteorite you observe is moving along a path
whose equation is 18x – y2 = -144. Will the meteorite strike Earth?
No. When 18x + 144 is substituted for y2 in the equation x2 + y2 = 40, the solutions of the resulting
equation are complex numbers.
Answers to Writing Exercises
60. x 2  bx Find the constant term.
2
1 2
1 
 b  b
4
2 
Add the constant term to the binomial to complete the square.
1
x 2  bx  b 2
4
Write the result as the square of a binomial.
1
x 2  bx  b 2
4
1 

  x  b
2 

2
Objective 8.3A
New Vocabulary
discriminant
New Formulas
quadratic formula
x
b  b 2  4ac
2a
Discuss the Concepts
1. If a quadratic equation is solved using the quadratic formula, and the result is x 
1  23
what are the
3
solutions of the equation?
26
, what are the
4
2. If a quadratic equation is solved using the quadratic formula, and the result is x 
solutions of the equation?
3. Suppose you must solve the quadratic equation x 2  3x  5. Does it matter whether you rewrite the
equation as x2  3x  5  0 or as 0   x2  3x  5 before you begin?
4. Which method of solving a quadratic equation do you prefer, completing the square or using the
quadratic formula? Why?
5. Explain why the discriminant determines whether a quadratic equation has one real number solution, two
real number solutions, or two complex number solutions.
Concept Check
1. For what value of k are the roots of 3x2  4x  k  0 equal?
4
3
2. Solve.
a. x 2  x 3  1  0
3 1
3 1
 i and 
 i
2 2
2 2
2
b. x  x 7  2  0


7 1
7 1
 i and 
 i
2 2
2 2
Optional Student Activity
1. For what values of k does the equation 2x2  kx  x  8  0 have a double root?
2. Find the difference between the larger root and the smaller root of x  px 
2
p
-7, 9
2
 1
4
 0.
3. What is the smallest integral value of K such that 2 x  Kx  4  x2  6  0 has no real roots?
1
2
4. Show that the equation x 2  bx  2  0 always has real number solutions regardless of the value of b.
b2 – 4ac = b2 – 4(1)(-2)
= b2 + 8
2
Because b is always greater than zero, b2 + 8 is always greater than zero. Therefore, x2 + bx – 2 has real
number solutions regardless of the value of b.
Answers to Writing Exercises
b  b 2  4ac
. In this formula a is the coefficient of x 2 , b is the coefficient
2a
of x, and c is the constant term in the quadratic equation ax2  bx  c  0 , a ≠ 0.
1. The quadratic formula is x 
2. The expression b2  4ac appears under the radical symbol in the quadratic formula. It is called the
discriminant. It can be used to determine whether a quadratic equation has one real number solution, two
unequal real number solutions, or two complex number solutions.
Answers to Writing Exercises
44. Students might mention some of the following points.
Factoring
Advantage: It is the easiest of the three methods to use.
Disadvantage: Not every quadratic equation can be solved by factoring.
Completing the Square
Advantage: This method can be used to solve any quadratic equation.
Disadvantage: This method is long and sometimes difficult to use.
Quadratic Formula
Advantage: This method can be used to solve any quadratic equation.
Disadvantage: Simplification of the resulting expression is sometimes difficult, and the quadratic
formula must be memorized.
Objective 8.4A
Vocabulary to Review
quadratic in form
[5.6C]
Discuss the Concepts
1. What does it mean for an equation to be quadratic in form?
2. Explain how to show that x 4  3x2  3  0 is quadratic in form.
3. Is the equation t 6  4t 3  4  0 quadratic in form? Why or why not?
4. Why does the equation x 2  5x  6  0 have two solutions while the equation x  5 x  6  0 has only
one solution?
Concept Check
1. Write two equations that are not quadratic equations but that can be written in quadratic form. Then write
them in quadratic form.
2. Solve: x2 / 5  6  5x1/ 5
32, 243
3. Solve:
x4
5x2
1 
4
4
-1, 1, -2, 2
4. Solve:
x4 8x2

3
3
3
-3, 3, -i, i
Optional Student Activity
 
2x
x
1. Solve: 2  32  12 2
2, 3
2. If 52 x  12  5x and 5x  2  N, find the value of N.
75
Objective 8.4B
Discuss the Concepts
1. What does the Property of Raising Each Side of an Equation to a Power state?
2. Is the equation x  3 x  8  0 quadratic in form? Explain why or why not.
3. Is the equation
4
x  2 3x  8  0 quadratic in form? Explain why or why not.
Concept Check
Determine whether the statement is always true, sometimes true, or never true.
1. An equation that is quadratic in form can be solved by using the quadratic formula.
2. Squaring both sides of a radical equation produces an extraneous root.
Always true
Sometimes true
3. When the Property of Raising Both Sides of an Equation to a Power is used to solve an equation, the
solutions must be checked.
Always true
Optional Student Activity
[Note: Exercises 1 and 2 require use of the Pythagorean Theorem. Exercises 3 and 4 require use of the
distance formula, d 
 x1  x2 
2
  y1  y2  .]
2
1. The width of a rectangle is twice the square root of the length. A diagonal of the rectangle is 12 in. Find
the length of the rectangle. Round to the nearest hundredth.
10.17 in.
2. The longer leg of a right triangle is four times the square root of the shorter leg. Find the lengths of the
two legs if the hypotenuse is 6 ft. Round to the nearest hundredth.
2 ft and 5.66 ft
3. Find two points on the line y  4 that are 5 units from the point  3,1 .
(-1, 4), (7, 4)
4. Find two points on the line y  6 that are 7 units from the point  5, 2  . Give both the exact values and
the approximations to the nearest hundredth.
5 



33, 6 and 5  33, 6 , or  0.74,6 and 10.74, 6
5. The total time from the instant you drop a rock into an abandoned mine shaft until the moment you hear
its impact is 4.5 s. Find the depth of the mine shaft. Use the equation 1100T  275 s  s , where T is the
time in seconds and s is the depth of the mine shaft in feet. Round to the nearest tenth.
287.5 ft
Objective 8.4C
Concept Check
1. For what values of x is the equation x 
1
1
 5  true?
x
5
5,
1
5
2. The reciprocal of x  1 is equal to x  1. Find the value of x.
3. Solve:
2 x 3 x
  
x 2 x 3
2,  2
6,  6
Optional Student Activity
1. Find the values of k such that the line containing the points (-2, -4) and  2  k,k  5 is parallel to the line
containing the points (1, -4) and  k,k  3 .
1
 ,3
2
2. A chemical reaction between carbon monoxide and water vapor is used to increase the concentration of
hydrogen gas in certain gas mixtures. In the process, carbon dioxide is also formed. For a certain
reaction, the concentration of carbon dioxide, x, in moles per liter is given by the equation
0.58 
x2
 0.02  x 
2
.
Solve this equation for x. Round to the nearest ten-thousandth.
0.0086 mole per liter
Objective 8.5A
New Vocabulary
quadratic inequality
Discuss the Concepts
1. For what values of x is x  5 positive? For what values is it negative?
2. Is 4 an element in the solution set of x2 – 6x + 5 > 0?
3. How does the solution set of x2 – 3x – 10 > 0 differ from the solution set of x2 – 3x – 10 ≥ 0?
Concept Check
Determine whether the statement is always true, sometimes true, or never true.
1. The endpoints of the solution set of a quadratic inequality are not included in the solution set.
Sometimes true
2. The solution set of (x – 3)(x – 2)(x + 2) > 0 is 3, 2, and -2.
Never true
3. The solution set of a quadratic inequality is an infinite set.
Sometimes true
4. The solution set of
1
 0 is {x| x > 1}.
x 1
Always true
Concept Check
1. How many integers satisfy the inequality x2 + 48 < 16x?
2. Find all values of x that satisfy both
x2
– 8 ≤ 2x and
Optional Student Activity
1. Solve: x < x2
{x| x < 0 or x > 1}
2. Solve: x3 > x
{x| -1 < x < 0 or x > 1}
3. Solve: (x –
4)2
> -2
All real numbers
x2
Seven
– 2x ≥ 8.
-2, 4
4. You shoot an arrow into the air with an initial velocity of 70 m/s. The distance up, in meters, is given by
d  rt  5t 2 , where t is the number of seconds since the arrow was shot and r is the initial velocity of the
arrow. Find the interval of time when the arrow will be more than 200 m high.
4 < t < 10; between 4 s and 10 s
Answers to Writing Exercises
1. It must be true that x – 3 > 0 and x – 5 > 0 or that x – 3 < 0 and x – 5 < 0. In other words, either both
factors are positive or both factors are negative.
2. 3 is not a possible element of the solution set because the value 3 makes the denominator equal to 0.
Objective 8.6A
Optional Student Activity
1. The sum of a number and twice its reciprocal is
33
. Find the number.
4
2. Find two consecutive integers whose cubes differ by 127.
1
or 8
4
6 and 7 or -6 and -7
3. The Tour de France is a bicycle race that begins in Dublin, Ireland, but is primarily a race through
France. On each day, racers cover a certain distance depending on the steepness of the terrain.
Suppose that on a particular day the racers must complete 210 mi. One cyclist, traveling 10 mph faster
than a second cyclist, covers this distance in 2.4 h less time than the second cyclist. Find the rate of the
first cyclist. 35 mph
4. A penalty kick in soccer is made from a penalty mark that is 36 ft from a goal that is 8 ft high. A possible
equation for the flight of a penalty kick is h  0.002x2  0.35x , where h is the height, in feet, of the ball x
feet from the penalty mark. Assuming that the flight of the kick is toward the goal and that the ball is not
touched by the goalie, will the ball land in the net? No
Answers to Focus on Problem Solving: Inductive and Deductive Reasoning
1. 82  64
2.
3. 14,386 is divisible by 2.
4. a. P, 62, 63
b. Inductive reasoning
5. a.
b. Inductive reasoning
6. a. 8  ’s  10  ‘s
b. Deductive reasoning
7. a. 3 ♥’s  6 ♠’s
b. Deductive reasoning
8. Inductive reasoning
9. Deductive reasoning
10. Deductive reasoning
Answers to Projects and Group Activities: Using a Graphing Calculator to
Solve a Quadratic Equation
1. 5.828427 ,  0.1715729
2. 1.302776 , 2.3027756
3. 2.224745, 0.22474487
4. No real number solution