9-5 Solving Quadratic Equations by Using the Quadratic Formula
The solutions are 2.3 and –5.3.
Solve each equation by using the Quadratic
Formula. Round to the nearest tenth if
necessary.
2
2. x − 10x + 16 = 0
2
6. 5x + 5 = −13x
SOLUTION: Write the equation in standard form.
SOLUTION: For this equation, a = 1, b = –10, and c = 16.
For this equation, a = 5, b = 13, and c = 5.
The solutions are 8 and 2.
2
4. x + 3x = 12
SOLUTION: Write the equation in standard form.
For this equation, a = 1, b = 3, and c = –12.
The solutions are –0.5 and –2.1.
Solve each equation. State which method you
used.
2
8. 2x − 3x − 6 = 0
SOLUTION: Solve using the quadratic formula.
For this equation, a = 2, b = –3, and c = –6.
Quadratic Formula:
The solutions are 2.3 and –5.3.
2
6. 5x + 5 = −13x
SOLUTION: Write the equation in standard form.
The solutions are 2.6 and –1.1.
2
10. x − 9x = −19
For this equation, a = 5, b = 13, and c = 5.
SOLUTION: Solve by using the quadratic formula.
Write the equation in standard form.
For this equation, a = 1, b = –9, and c = 19.
Quadratic Formula:
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The solutions are –0.5 and –2.1.
Page 1
The discriminant is 41.
Since the discriminant is positive, the equation has
9-5 Solving
Quadratic
The solutions
are 2.6Equations
and –1.1. by Using the Quadratic Formula
two real solutions.
2
10. x − 9x = −19
SOLUTION: Solve by using the quadratic formula.
Write the equation in standard form.
2
14. 3x − x = 8
SOLUTION: Write the equation in standard form.
For this equation, a = 3, b = –1, and c = –8.
For this equation, a = 1, b = –9, and c = 19.
Quadratic Formula:
The discriminant is 97.
Since the discriminant is positive, the equation has
two real solutions.
Solve each equation by using the Quadratic
Formula. Round to the nearest tenth if
necessary.
2
16. 4x + 5x − 6 = 0
SOLUTION: For this equation, a = 4, b = 5, and c = –6.
The solutions are 5.6 and 3.4.
State the value of the discriminant for each
equation. Then determine the number of real
solutions of the equation.
2
12. 2x − 11x + 10 = 0
SOLUTION: For this equation, a = 2, b = –11, and c = 10.
The solutions are
The discriminant is 41.
Since the discriminant is positive, the equation has
two real solutions.
2
and –2.
2
18. 6x − 12x + 1 = 0
SOLUTION: For this equation, a = 6, b = –12, and c = 1.
14. 3x − x = 8
SOLUTION: Write the equation in standard form.
For this equation, a = 3, b = –1, and c = –8.
The solutions are 1.9 and 0.1.
The Manual
discriminant
is 97.
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Since the discriminant is positive, the equation has
two real solutions.
2
20. 2x − 5x = −7
SOLUTION: Write the equation in standard form.
Page 2
9-5 Solving Quadratic Equations by Using the Quadratic Formula
The solutions are
The solutions are 1.9 and 0.1.
.
2
2
20. 2x − 5x = −7
24. 4x = −16x − 16
SOLUTION: Write the equation in standard form.
SOLUTION: Write the equation in standard form.
For this equation, a = 2, b = –5, and c = 7.
For this equation, a = 4, b = 16, and c = 16.
The discriminant is negative, so the equation has no
real solutions, Ø.
2
22. 81x = 9
The solution is –2.
SOLUTION: Write the equation in standard form.
2
26. −3x = 8x − 12
SOLUTION: Write the equation in standard form.
For this equation, a = 81, b = 0, and c = –9.
For this equation, a = –3, b = –8, and c = 12.
The solutions are
.
2
24. 4x = −16x − 16
SOLUTION: Write the equation in standard form.
For this equation, a = 4, b = 16, and c = 16.
The solutions are –3.7 and 1.1.
28. AMUSEMENT PARKS The Demon Drop at
Cedar Point in Ohio takes riders to the top of a tower
and drops them vertically 60 feet. A function that
2
approximates this drop is h = −16t + 60, where h is
the height in feet and t is the time in seconds. About
how many seconds does it take for riders to drop 60
feet?
SOLUTION: 2
−16t + 60 = 0
For this equation, a = –16, b = 0, and c = 60.
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It takes about 1.9 seconds for the riders to drop 60
9-5 Solving Quadratic Equations by Using the Quadratic Formula
feet.
The solutions are –3.7 and 1.1.
28. AMUSEMENT PARKS The Demon Drop at
Cedar Point in Ohio takes riders to the top of a tower
and drops them vertically 60 feet. A function that
2
approximates this drop is h = −16t + 60, where h is
the height in feet and t is the time in seconds. About
how many seconds does it take for riders to drop 60
feet?
Solve each equation. State which method you
used.
2
30. 3x − 24x = −36
SOLUTION: Solve by using the quadratic formula.
Write the equation in standard form.
SOLUTION: 2
−16t + 60 = 0
For this equation, a = –16, b = 0, and c = 60.
For this equation, a = 3, b = –24, and c = 36.
Use the Quadratic Formula.
The solutions are 6 and 2.
2
32. 4x + 100 = 0
It takes about 1.9 seconds for the riders to drop 60
feet.
Solve each equation. State which method you
used.
2
30. 3x − 24x = −36
SOLUTION: Solve by using the quadratic formula.
Write the equation in standard form.
For this equation, a = 3, b = –24, and c = 36.
Use the Quadratic Formula.
SOLUTION: Solve using the quadratic formula.
First, check the value of the discriminant.
For this equation, a = 4, b = 0, and c = 100.
The discriminant is negative, so the equation has no
real solutions, ø.
34. 12 − 12x = −3x
2
SOLUTION: Solve by factoring.
Write the equation in standard form.
Factor.
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Page 4
For this equation, a = 4, b = 0, and c = 100.
The discriminant is 0.
The discriminant
is negative,
so by
theUsing
equation
no
Since the discriminant is 0, the equation has one real
9-5 Solving
Quadratic
Equations
thehas
Quadratic
Formula
solution.
real solutions, ø.
34. 12 − 12x = −3x
2
SOLUTION: Solve by factoring.
Write the equation in standard form.
40. SOLUTION: Write the equation in standard form.
Factor.
For this equation, a = 2, b =
, and c =
.
The solution is 2.
State the value of the discriminant for each
equation. Then determine the number of real
solutions of the equation.
2
36. 2x − 5x + 20 = 0
SOLUTION: For this equation, a = 2, b = –5, and c = 20.
The discriminant is –135.
Since the discriminant is negative, the equation has
no real solutions.
The discriminant is 18.25.
Since the discriminant is positive, the equation has
two real solutions.
2
42. TRAFFIC The equation d = 0.05v + 1.1v models
the distance d in feet it takes a car traveling at a
speed of v miles per hour to come to a complete
stop. If Hannah’s car stopped after 250 feet on a
highway with a speed limit of 65 miles per hour, was
she speeding? Explain your reasoning.
SOLUTION: Write the equation in standard form.
2
38. 0.5x − 2x = −2
SOLUTION: Write the equation in standard form.
For this equation, a = 0.05, b = 1.1, and c = –250.
For this equation, a = 0.5, b = –2, and c = 2.
The discriminant is 0.
Since the discriminant is 0, the equation has one real
solution.
40. SOLUTION: Write the equation in standard form.
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No, she was not speeding; Sample answer: Hannah
was traveling at about 61 mph, so she was not
speeding.
Without graphing, determine the number ofPage
x- 5
intercepts of the graph of the related function
for each function.
The discriminant is 0.04.
No, she was not speeding; Sample answer: Hannah
Since the discriminant is positive, the graph of the
9-5 Solving
Quadratic
Equations
by she
Using
was traveling
at about
61 mph, so
wasthe
notQuadratic Formula
function will have two x-intercepts.
speeding.
Without graphing, determine the number of xintercepts of the graph of the related function
for each function.
44. Solve each equation by using the Quadratic
Formula. Round to the nearest tenth if
necessary.
2
46. −2x − 7x = −1.5
SOLUTION: Write the equation in standard form.
SOLUTION: Write the equation in standard form.
For this equation, a = –2, b = –7, and c = 1.5.
For this equation, a = 1, b =
, and c =
.
The discriminant is 0.04.
Since the discriminant is positive, the graph of the
function will have two x-intercepts.
Solve each equation by using the Quadratic
Formula. Round to the nearest tenth if
necessary.
2
46. −2x − 7x = −1.5
SOLUTION: Write the equation in standard form.
The solutions are –3.7 and 0.2.
2
48. x − 2x = 5
SOLUTION: Write the equation in standard form.
For this equation, a = 1, b = –2, and c = –5.
For this equation, a = –2, b = –7, and c = 1.5.
The solutions are 3.4 and –1.4.
The solutions are –3.7 and 0.2.
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2
48. x − 2x = 5
SOLUTION: 50. MULTIPLE REPRESENTATIONS In this
problem, you will investigate writing a quadratic Page 6
equation with given roots. 2
If p is a root of 0 = ax + bx + c, then (x – p ) is a
2
c. You could write an equation with three roots by
multiplying the corresponding factors together and
setting it equal to zero. If an equation has the three
roots 1, 2, 3, then the corresponding factors would be
The solutions
are 3.4Equations
and –1.4. by Using the Quadratic Formula
9-5 Solving
Quadratic
(x – 1), (x – 2), and (x – 3). The equation would then
be:
50. MULTIPLE REPRESENTATIONS In this
problem, you will investigate writing a quadratic
equation with given roots. 2
If p is a root of 0 = ax + bx + c, then (x – p ) is a
2
factor of ax + bx + c.
This is not a quadratic equation since it is of degree
3. a. Tabular Copy and complete the first two columns
of the table.
b. Algebraic Multiply the factors to write each
equation with integral coefficients. Use the equations
to complete the last column of the table. Write each
equation.
c. Analytical How could you write an equation with
three roots? Test your conjecture by writing an
equation with roots 1, 2, and 3. Is the equation
quadratic? Explain.
SOLUTION: a. For any two roots m and p , in the left hand column,
the middle column will be the corresponding factors
(x – m), (x – p ). b. The equation with these factors will be: (x – m)(x
2
– p ) = 0 which simplifies to x – (m + p )x + mp = 0.
Use this to fill in the column of the table. 52. REASONING Use factoring techniques to
2
determine the number of real zeros of f (x) = x − 8x
+ 16. Compare this method to using the discriminant.
SOLUTION: 2
For f(x) = x − 8x + 16, a = 1, b = −8 and c = 16.
2
2
Then the discriminate is b − 4ac or (−8) −4(1)(16)
= 0. The polynomial can be factored to get f (x) = (x
2.
− 4) Solve to find the real zeros.
So the only real zero is 4. The discriminant is 0, so
the only real zero is 4. The discriminant is 0, so there
is 1 real zero. The discriminant tells us how many
real zeros there are. Factoring tells us what they are.
CCSS STRUCTURE Determine whether there
are two, one, or no real solutions.
54. The graph of a quadratic function touches but does
not cross the x-axis.
SOLUTION: If the graph is tangent to the x-axis, meaning there is
only one x-intercept, then there is only one real
solution.
c. You could write an equation with three roots by
multiplying the corresponding factors together and
setting it equal to zero. If an equation has the three
roots 1, 2, 3, then the corresponding factors would be
(x – 1), (x – 2), and (x – 3). The equation would then
be:
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56. Both a and b are greater than 0 and c is less than 0
in a quadratic equation.
SOLUTION: 2
The discrimininant is b – 4ac. No matter the value
2
of b, b will always be positive. If a is greater than 0
and c is less than 0, then – 4ac will be positive. Thus
Page 7
the discrimininant would be positive. So there would be two real solutions.
leading coefficient has to be 1 and the x - and xSOLUTION: term must be isolated. It is also easier if the
If the graph is tangent to the x-axis, meaning there is
coefficient of the x-term is even; if not, the
only one x-intercept, then there is only one real
calculations become harder when dealing with
9-5 Solving
2
solution.Quadratic Equations by Using the Quadratic Formula
fractions. For example x + 4x = 7 can be solved by
completing the square. 56. Both a and b are greater than 0 and c is less than 0
in a quadratic equation.
SOLUTION: 2
The discrimininant is b – 4ac. No matter the value
2
of b, b will always be positive. If a is greater than 0
and c is less than 0, then – 4ac will be positive. Thus
the discrimininant would be positive. So there would be two real solutions.
58. WRITING IN MATH Describe the advantages
and disadvantages of each method of solving
quadratic equations. Which method do you prefer,
and why?
Quadratic Formula:
The Quadratic Formula will work for any quadratic
equation and exact solutions can be found. This
method can be time consuming, especially if an
equation is easily factored. For example, use the 2
Quadratic Formula to find the solutions of f (x) = 4x
+ 13 x + 5. SOLUTION: Factoring:
Factoring is easy if the polynomial is factorable and
complicated if it is not. Not all equations are
factorable. 2
2
For example f (x) = x – 8x + 16 factors to (x – 4) .
2
However, f (x) = x – 16x + 8 can not be factored.
Graphing:
Graphing only gives approximate answers, but it is
easy to see the number of solutions. Using square
roots is easy when there is no x-term. See students’ preferences.
60. SHORT RESPONSE The triangle shown is an
isosceles triangle. What is the value of x?
2
For example, for the quadratic f (x) = 2x – 17x + 4,
you can see the two solutions in the graph. However,
it will be difficult to identify the solution x =
8.2578049 in the graph. SOLUTION: Because an isosceles triangle has two equal angles, x
could be equal to 64, or it could be equal to the
unnamed angle, where 180 = 2x + 64.
[-5, 15] scl: 2 by [-30, 10] scl: 4
The value of x is 58 or 64.
Completing the square:
Completing the square can be used for any quadratic
equation and exact solutions can be found, but the
2
leading coefficient has to be 1 and the x - and xterm must be isolated. It is also easier if the
coefficient of the x-term is even; if not, the
calculations become harder when dealing with
2
fractions. For example x + 4x = 7 can be solved by
completing the square. 62. What are the solutions of the quadratic equation 6h
+ 6h = 72?
A 3 or −4
B −3 or 4
C no solution
D 12 or −48
2
SOLUTION: Write the equation in standard form.
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Page 8
For this equation, a = 6, b = 6, and c = –72.
9-5 Solving Quadratic Equations by Using the Quadratic Formula
The solutions are 1.6 or 7.4.
The value of x is 58 or 64.
62. What are the solutions of the quadratic equation 6h
+ 6h = 72?
A 3 or −4
B −3 or 4
C no solution
D 12 or −48
2
Describe the transformations needed to obtain
the graph of g(x) from the graph of f (x).
2
66. f (x) = 4x
2
g(x) = 2x
SOLUTION: 2
The graph of g(x) = ax stretches or compresses the
SOLUTION: Write the equation in standard form.
2
graph of f (x) = 4x vertically. The change in a is
< 1. If 0 < and 0 <
For this equation, a = 6, b = 6, and c = –72.
< 1, the graph of f (x) = x
,
2
is compressed vertically. Therefore, the graph of y =
2
2
2x is the graph of y = 4x vertically compressed.
2
68. f (x) = x − 6
2
g(x) = x + 3
SOLUTION: 2
The graph of f (x) = x + c represents a vertical
translation of the parent graph. The value of the
change in c is 9, and 9 > 0. If c > 0, the graph of f (x)
2
= x is translated units up. Therefore, the graph of
2
2
y = x +3 is a translation of the graph of y = x –6
shifted up 9 units.
Determine whether each graph shows a positive
correlation, a negative correlation, or no
correlation. If there is a positive or negative
correlation, describe its meaning in the
situation.
The solutions are –4 or 3.
So, the correct choicer is A.
Solve each equation by completing the square.
Round to the nearest tenth if necessary.
2
64. x − 9x = −12
SOLUTION: 70. The solutions are 1.6 or 7.4.
Describe the transformations needed to obtain
the graph of g(x) from the graph of f (x).
2
66. f (x) = 4x
2
g(x) = 2x
SOLUTION: 2
The graph of g(x) = ax stretches or compresses the
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graph of f (x) = 4x vertically. The change in a is
,
2
SOLUTION: The graph shows no correlation between the year
and the number of hurricanes because the points are
randomly spread out.
Determine whether each sequence is
arithmetic, geometric, or neither. Explain.
72. 20, 25, 30, ...
Page 9
SOLUTION: Check the difference and ratio between terms. SOLUTION: The graph shows no correlation between the year
and the number of hurricanes because the points are
9-5 Solving
Quadratic
randomly
spread out.Equations by Using the Quadratic Formula
Determine whether each sequence is
arithmetic, geometric, or neither. Explain.
72. 20, 25, 30, ...
SOLUTION: Check the difference and ratio between terms. 25 – 20 = 5
30 – 25 = 5
There is a common difference of 5 between the
terms. The sequence is arithmetic.
74. 200, 350, 650, …
SOLUTION: Check the difference and ratio between terms. 350 – 200 = 150
650 – 350 = 200
There is no common difference between the terms. There is no common ratio. This is neither an
arithmetic nor geometric sequence. 76. 2, 4, 16, …
SOLUTION: Check the difference and ratio between terms. 4–2=2
16 – 4 = 12
There is no common difference between the terms. There is no common difference or ratio between the
terms. This is neither a geometric nor an arithmetic
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