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Algebra 2 and Trigonometry

Ann Xavier Gantert

Chapter 5

QUADRATIC FUNCTIONS AND COMPLEX NUMBERS

Educators


Problem 1

To solve the equation given in step 2 of Example 2 , Sarah multiplied each side of the equation by 8 and then added 1 to each side to complete the square. Show that $16 x^{2}-8 x-8$ is the square of a binomial and will lead to the correct solution of $0=2 x^{2}-x-1$

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Problem 2

Phillip said that the equation $0=x^{2}-6 x+1$ can be solved by adding 8 to both sides of the equation. Do you agree with Phillip? Explain why or why not.

Aditya B.
Numerade Educator

Problem 3

In $3-8,$ complete the square of the quadratic expression.
$$
x^{2}+6 x
$$

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Problem 4

In $3-8,$ complete the square of the quadratic expression.
$$
x^{2}-8 x
$$

Aditya B.
Numerade Educator

Problem 5

In $3-8,$ complete the square of the quadratic expression.
$$
x^{2}-2 x
$$

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Problem 6

In $3-8,$ complete the square of the quadratic expression.
$$
x^{2}-12 x
$$

Aditya B.
Numerade Educator

Problem 7

In $3-8,$ complete the square of the quadratic expression.
$$
2 x^{2}-4 x
$$

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Problem 8

In $3-8,$ complete the square of the quadratic expression.
$$
x^{2}-3 x
$$

Aditya B.
Numerade Educator

Problem 9

In $9-14 :$ a Sketch the graph of each function. b. From the graph, estimate the roots of the function to the nearest tenth. c. Find the exact irrational roots in simplest radical form.
$$
\mathrm{f}(x)=x^{2}-6 x+4
$$

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Problem 10

In $9-14 :$ a Sketch the graph of each function. b. From the graph, estimate the roots of the function to the nearest tenth. c. Find the exact irrational roots in simplest radical form.
$$
\mathrm{f}(x)=x^{2}-2 x-2
$$

Aditya B.
Numerade Educator

Problem 11

In $9-14 :$ a Sketch the graph of each function. b. From the graph, estimate the roots of the function to the nearest tenth. c. Find the exact irrational roots in simplest radical form.
$$
f(x)=x^{2}+4 x+2
$$

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Problem 12

In $9-14 :$ a Sketch the graph of each function. b. From the graph, estimate the roots of the function to the nearest tenth. c. Find the exact irrational roots in simplest radical form.
$$
f(x)=x^{2}-6 x+6
$$

Aditya B.
Numerade Educator

Problem 13

In $9-14 :$ a Sketch the graph of each function. b. From the graph, estimate the roots of the function to the nearest tenth. c. Find the exact irrational roots in simplest radical form.
$$
f(x)=x^{2}-2 x-1
$$

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Problem 14

In $9-14 :$ a Sketch the graph of each function. b. From the graph, estimate the roots of the function to the nearest tenth. c. Find the exact irrational roots in simplest radical form.
$$
f(x)=x^{2}-10 x+18
$$

Aditya B.
Numerade Educator

Problem 15

In $9-26,$ solve each quadratic equation by completing the square. Express the answer in simplest radical form.
$$
x^{2}-2 x-2=0
$$

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Problem 16

In $9-26,$ solve each quadratic equation by completing the square. Express the answer in simplest radical form.
$$
x^{2}+6 x+4=0
$$

Aditya B.
Numerade Educator

Problem 17

In $9-26,$ solve each quadratic equation by completing the square. Express the answer in simplest radical form.
$$
x^{2}-4 x+1=0
$$

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Problem 18

In $9-26,$ solve each quadratic equation by completing the square. Express the answer in simplest radical form.
$$
x^{2}+2 x-5=0
$$

Aditya B.
Numerade Educator

Problem 19

In $9-26,$ solve each quadratic equation by completing the square. Express the answer in simplest radical form.
$$
x^{2}-6 x+2=0
$$

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Problem 20

In $9-26,$ solve each quadratic equation by completing the square. Express the answer in simplest radical form.
$$
x^{2}-8 x+4=0
$$

Aditya B.
Numerade Educator

Problem 21

In $9-26,$ solve each quadratic equation by completing the square. Express the answer in simplest radical form.
$$
2 x^{2}+12 x+3=0
$$

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Problem 22

In $9-26,$ solve each quadratic equation by completing the square. Express the answer in simplest radical form.
$$
2 x^{2}-6 x+3=0
$$

Aditya B.
Numerade Educator

Problem 23

In $9-26,$ solve each quadratic equation by completing the square. Express the answer in simplest radical form.
$$
2 x^{2}-6 x+3=0
$$

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Problem 24

In $9-26,$ solve each quadratic equation by completing the square. Express the answer in simplest radical form.
$$
4 x^{2}-20 x+9=0
$$

Aditya B.
Numerade Educator

Problem 25

In $9-26,$ solve each quadratic equation by completing the square. Express the answer in simplest radical form.
$$
\frac{1}{2} x^{2}+x-3=0
$$

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Problem 26

In $9-26,$ solve each quadratic equation by completing the square. Express the answer in simplest radical form.
$$
\frac{1}{4} x^{2}+\frac{3}{4} x-\frac{3}{2}=0
$$

Aditya B.
Numerade Educator

Problem 27

a. Complete the square to find the roots of the equation $x^{2}-5 x+1=0$ .
b. Write, to the nearest tenth, a rational approximation for the roots.

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Problem 28

In $28-33,$ without graphing the parabola, describe the translation, reflection, and $/$ or scaling that must be applied to $y=x^{2}$ to obtain the graph of each given function.
$$
\mathrm{f}(x)=x^{2}-12 x+5
$$

Aditya B.
Numerade Educator

Problem 29

In $28-33,$ without graphing the parabola, describe the translation, reflection, and $/$ or scaling that must be applied to $y=x^{2}$ to obtain the graph of each given function.
$$
\mathrm{f}(x)=x^{2}+2 x-2
$$

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Problem 30

In $28-33,$ without graphing the parabola, describe the translation, reflection, and $/$ or scaling that must be applied to $y=x^{2}$ to obtain the graph of each given function.
$$
f(x)=x^{2}-6 x-7
$$

Aditya B.
Numerade Educator

Problem 31

In $28-33,$ without graphing the parabola, describe the translation, reflection, and $/$ or scaling that must be applied to $y=x^{2}$ to obtain the graph of each given function.
$$
\mathrm{f}(x)=x^{2}+x+\frac{9}{4}
$$

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Problem 32

In $28-33,$ without graphing the parabola, describe the translation, reflection, and $/$ or scaling that must be applied to $y=x^{2}$ to obtain the graph of each given function.
$$
\mathrm{f}(x)=-x^{2}+x+2
$$

Aditya B.
Numerade Educator

Problem 33

In $28-33,$ without graphing the parabola, describe the translation, reflection, and $/$ or scaling that must be applied to $y=x^{2}$ to obtain the graph of each given function.
$$
f(x)=3 x^{2}+6 x+3
$$

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Problem 34

Determine the coordinates of the vertex and the equation of the axis of symmetry of $\mathrm{f}(x)=x^{2}+8 x+5$ by writing the equation in the form $\mathrm{f}(x)=(x-h)^{2}+k .$ Justify your answer.

Aditya B.
Numerade Educator

Problem 35

The length of a rectangle is 4 feet more than twice the width. The area of the rectangle is 38 square feet.
a. Find the dimensions of the rectangle in simplest radical form.
b. Show that the product of the length and width is equal to the area.
c. Write, to the nearest tenth, rational approximations for the length and width.

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Problem 36

One base of a trapezoid is 8 feet longer than the other base and the height of the trapezoid is equal to the length of the shorter base. The area of the trapezoid is 20 square feet.
a. Find the lengths of the bases and of the height of the trapezoid in simplest radical form.
b. Show the lengths of the basezoid is equal to one-half the height times the sum of the lengths of the bases.
c. Write, to the nearest tenth, rational approximations for the lengths of the bases and for the height of the trapezoid.

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Problem 37

Steve and Alice realized that their ages are consecutive odd integers. The product of their ages is $195 .$ Steve is younger than Alice. Determine their ages by completing the square.

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