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Beginning and Intermediate Algebra

Julie Miller,Molly O’Neill,Nancy Hyde

Chapter 11

Quadratic Equations and Functions - all with Video Answers

Educators

Section 1

Square Root Property and Completing the Square

02:00

Problem 1

Define the key terms.
a. square root property
b. completing the square

Narayan Hari
Narayan Hari
Numerade Educator
01:01

Problem 2

Solve the equations by using the square root property. (See Examples $1-3 .$ )
$$x^{2}=100$$

Narayan Hari
Narayan Hari
Numerade Educator
01:01

Problem 3

Solve the equations by using the square root property. (See Examples $1-3 .$ )
$$y^{2}=4$$

Narayan Hari
Narayan Hari
Numerade Educator
01:04

Problem 4

Solve the equations by using the square root property. (See Examples $1-3 .$ )
$$a^{2}=5$$

Narayan Hari
Narayan Hari
Numerade Educator
01:01

Problem 5

Solve the equations by using the square root property. (See Examples $1-3 .$ )
$$k^{2}-7=0$$

Narayan Hari
Narayan Hari
Numerade Educator
01:05

Problem 6

Solve the equations by using the square root property. (See Examples $1-3 .$ )
$$3 v^{2}+33=0$$

Narayan Hari
Narayan Hari
Numerade Educator
01:04

Problem 7

Solve the equations by using the square root property. (See Examples $1-3 .$ )
$$-2 m^{2}=50$$

Narayan Hari
Narayan Hari
Numerade Educator
01:00

Problem 8

Solve the equations by using the square root property. (See Examples $1-3 .$ )
$$(p-5)^{2}=9$$

Narayan Hari
Narayan Hari
Numerade Educator
01:00

Problem 9

Solve the equations by using the square root property. (See Examples $1-3 .$ )
$$(q+3)^{2}=4$$

Narayan Hari
Narayan Hari
Numerade Educator
01:05

Problem 10

Solve the equations by using the square root property. (See Examples $1-3 .$ )
$$(3 x-2)^{2}-5=0$$

Narayan Hari
Narayan Hari
Numerade Educator
01:07

Problem 11

Solve the equations by using the square root property. (See Examples $1-3 .$ )
$$(2 y+3)^{2}-7=0$$

Narayan Hari
Narayan Hari
Numerade Educator
01:09

Problem 12

Solve the equations by using the square root property. (See Examples $1-3 .$ )
$$(h-4)^{2}=-8$$

Narayan Hari
Narayan Hari
Numerade Educator
01:07

Problem 13

Solve the equations by using the square root property. (See Examples $1-3 .$ )
$$(t+5)^{2}=-18$$

Narayan Hari
Narayan Hari
Numerade Educator
01:01

Problem 14

Solve the equations by using the square root property. (See Examples $1-3 .$ )
$$6 p^{2}-3=2$$

Narayan Hari
Narayan Hari
Numerade Educator
02:46

Problem 15

Solve the equations by using the square root property. (See Examples $1-3 .$ )
$$15=4+3 w^{2}$$

P Krishnamurthy
P Krishnamurthy
Numerade Educator
01:07

Problem 16

Solve the equations by using the square root property. (See Examples $1-3 .$ )
$$\left(x-\frac{3}{2}\right)^{2}+\frac{7}{4}=0$$

Narayan Hari
Narayan Hari
Numerade Educator
01:25

Problem 17

Solve the equations by using the square root property. (See Examples $1-3 .$ )
$$\left(m+\frac{4}{5}\right)^{2}+\frac{3}{25}=0$$

Narayan Hari
Narayan Hari
Numerade Educator
00:29

Problem 18

Given the equation $x^{2}=k,$ match the following statements.
a. If $k>0,$ then ______
b. If $k<0,$ then ______
c. If $k=0,$ then ______
i. there will be one real solution.
ii. there will be two real solutions.
iii. there will be two imaginary solutions.

AG
Ankit Gupta
Numerade Educator
01:39

Problem 19

State two methods that can be used to solve the equation $x^{2}-36=0 .$ Then solve the equation by using both methods.

AG
Ankit Gupta
Numerade Educator
00:36

Problem 20

Explain the difference between solving the equations: $x=\sqrt{16}$ and $x^{2}=16$

AG
Ankit Gupta
Numerade Educator
01:01

Problem 21

Find the value of $n$ so that the expression is a perfect square trinomial. Then factor the trinomial. (See Example 4.)
$$x^{2}-6 x+n$$

Narayan Hari
Narayan Hari
Numerade Educator
01:02

Problem 22

Find the value of $n$ so that the expression is a perfect square trinomial. Then factor the trinomial. (See Example 4.)
$$x^{2}+12 x+n$$

Narayan Hari
Narayan Hari
Numerade Educator
01:00

Problem 23

Find the value of $n$ so that the expression is a perfect square trinomial. Then factor the trinomial. (See Example 4.)
$$t^{2}+8 t+n$$

Narayan Hari
Narayan Hari
Numerade Educator
01:01

Problem 24

Find the value of $n$ so that the expression is a perfect square trinomial. Then factor the trinomial. (See Example 4.)
$$v^{2}-18 v+n$$

Narayan Hari
Narayan Hari
Numerade Educator
01:01

Problem 25

Find the value of $n$ so that the expression is a perfect square trinomial. Then factor the trinomial. (See Example 4.)
$$c^{2}-c+n$$

Narayan Hari
Narayan Hari
Numerade Educator
01:02

Problem 26

Find the value of $n$ so that the expression is a perfect square trinomial. Then factor the trinomial. (See Example 4.)
$$x^{2}+9 x+n$$

Narayan Hari
Narayan Hari
Numerade Educator
01:01

Problem 27

Find the value of $n$ so that the expression is a perfect square trinomial. Then factor the trinomial. (See Example 4.)
$$y^{2}+5 y+n$$

Narayan Hari
Narayan Hari
Numerade Educator
01:01

Problem 28

Find the value of $n$ so that the expression is a perfect square trinomial. Then factor the trinomial. (See Example 4.)
$$a^{2}-7 a+n$$

Narayan Hari
Narayan Hari
Numerade Educator
01:09

Problem 29

Find the value of $n$ so that the expression is a perfect square trinomial. Then factor the trinomial. (See Example 4.)
$$b^{2}+\frac{2}{5} b+n$$

Narayan Hari
Narayan Hari
Numerade Educator
01:01

Problem 30

Find the value of $n$ so that the expression is a perfect square trinomial. Then factor the trinomial. (See Example 4.)
$$m^{2}-\frac{2}{7} m+n$$

Narayan Hari
Narayan Hari
Numerade Educator
01:01

Problem 31

Find the value of $n$ so that the expression is a perfect square trinomial. Then factor the trinomial. (See Example 4.)
$$p^{2}-\frac{2}{3} p+n$$

Narayan Hari
Narayan Hari
Numerade Educator
01:02

Problem 32

Find the value of $n$ so that the expression is a perfect square trinomial. Then factor the trinomial. (See Example 4.)
$$w^{2}+\frac{3}{4} w+n$$

Narayan Hari
Narayan Hari
Numerade Educator
01:47

Problem 33

Summarize the steps used in solving a quadratic equation by completing the square and applying the square root property.

Narayan Hari
Narayan Hari
Numerade Educator
00:39

Problem 34

What types of quadratic equations can be solved by completing the square and applying the square root property?

AG
Ankit Gupta
Numerade Educator
01:19

Problem 35

Solve the quadratic equation by completing the square and applying the square root property. (See Examples $5-7$ )
$$t^{2}+8 t+15=0$$

Narayan Hari
Narayan Hari
Numerade Educator
01:18

Problem 36

Solve the quadratic equation by completing the square and applying the square root property. (See Examples $5-7$ )
$$m^{2}+6 m+8=0$$

Narayan Hari
Narayan Hari
Numerade Educator
01:03

Problem 37

Solve the quadratic equation by completing the square and applying the square root property. (See Examples $5-7$ )
$$x^{2}+6 x=-16$$

Narayan Hari
Narayan Hari
Numerade Educator
04:32

Problem 38

Solve the quadratic equation by completing the square and applying the square root property. (See Examples $5-7$ )
$$x^{2}-4 x=-15$$

P Krishnamurthy
P Krishnamurthy
Numerade Educator
01:04

Problem 39

Solve the quadratic equation by completing the square and applying the square root property. (See Examples $5-7$ )
$$p^{2}+4 p+6=0$$

Narayan Hari
Narayan Hari
Numerade Educator
01:12

Problem 40

Solve the quadratic equation by completing the square and applying the square root property. (See Examples $5-7$ )
$$q^{2}+2 q+2=0$$

Narayan Hari
Narayan Hari
Numerade Educator
01:21

Problem 41

Solve the quadratic equation by completing the square and applying the square root property. (See Examples $5-7$ )
$$-3 y-10=-y^{2}$$

Narayan Hari
Narayan Hari
Numerade Educator
01:19

Problem 42

Solve the quadratic equation by completing the square and applying the square root property. (See Examples $5-7$ )
$$-24=-2 y^{2}+2 y$$

Narayan Hari
Narayan Hari
Numerade Educator
01:15

Problem 43

Solve the quadratic equation by completing the square and applying the square root property. (See Examples $5-7$ )
$$2 a^{2}+4 a+5=0$$

Narayan Hari
Narayan Hari
Numerade Educator
01:24

Problem 44

Solve the quadratic equation by completing the square and applying the square root property. (See Examples $5-7$ )
$$3 a^{2}+6 a-7=0$$

Narayan Hari
Narayan Hari
Numerade Educator
01:11

Problem 45

Solve the quadratic equation by completing the square and applying the square root property. (See Examples $5-7$ )
$$9 x^{2}-36 x+40=0$$

Narayan Hari
Narayan Hari
Numerade Educator
01:17

Problem 46

Solve the quadratic equation by completing the square and applying the square root property. (See Examples $5-7$ )
$$9 y^{2}-12 y+5=0$$

Narayan Hari
Narayan Hari
Numerade Educator
01:19

Problem 47

Solve the quadratic equation by completing the square and applying the square root property. (See Examples $5-7$ )
$$25 p^{2}-10 p=2$$

Narayan Hari
Narayan Hari
Numerade Educator
01:18

Problem 48

Solve the quadratic equation by completing the square and applying the square root property. (See Examples $5-7$ )
$$9 n^{2}-6 n=1$$

Narayan Hari
Narayan Hari
Numerade Educator
01:38

Problem 49

Solve the quadratic equation by completing the square and applying the square root property. (See Examples $5-7$ )
$$(2 w+5)(w-1)=2$$

Narayan Hari
Narayan Hari
Numerade Educator
01:24

Problem 50

Solve the quadratic equation by completing the square and applying the square root property. (See Examples $5-7$ )
$$(3 p-5)(p+1)=-3$$

Narayan Hari
Narayan Hari
Numerade Educator
01:01

Problem 51

Solve the quadratic equation by completing the square and applying the square root property. (See Examples $5-7$ )
$$n(n-4)=7$$

Narayan Hari
Narayan Hari
Numerade Educator
01:20

Problem 52

Solve the quadratic equation by completing the square and applying the square root property. (See Examples $5-7$ )
$$m(m+10)=2$$

Narayan Hari
Narayan Hari
Numerade Educator
01:22

Problem 53

Solve the quadratic equation by completing the square and applying the square root property. (See Examples $5-7$ )
$$2 x(x+6)=14$$

Narayan Hari
Narayan Hari
Numerade Educator
01:08

Problem 54

Solve the quadratic equation by completing the square and applying the square root property. (See Examples $5-7$ )
$$3 x(x-2)=24$$

Narayan Hari
Narayan Hari
Numerade Educator
01:01

Problem 55

The distance (in feet) that an object falls in $t$ sec is given by the equation $d=16 t^{2},$ where $t \geq 0$
a. Solve the equation for $t$ ( See Example 8.)
b. Using the equation from part (a), determine the amount of time required for an object to fall $1024 \mathrm{ft}$.

Narayan Hari
Narayan Hari
Numerade Educator
01:16

Problem 56

The volume of a can that is 4 in. tall is given by the equation $V=4 \pi r^{2},$ where $r$ is the radius of the can, measured in inches.
a. Solve the equation for $r$ Do not rationalize the denominator.
b. Using the equation from part (a), determine the radius of a can with volume of 12.56 in. $^{3}$ Use 3.14 for $\pi .$

Narayan Hari
Narayan Hari
Numerade Educator
01:03

Problem 57

Solve for the indicated variable.
$$A=\pi r^{2} \text { for } r \quad(r>0)$$

AG
Ankit Gupta
Numerade Educator
01:06

Problem 58

Solve for the indicated variable.
$$E=m c^{2} \text { for } c \quad(c>0)$$

AG
Ankit Gupta
Numerade Educator
01:13

Problem 59

Solve for the indicated variable.
$$a^{2}+b^{2}+c^{2}=d^{2} \quad \text { for } a \quad(a>0)$$

AG
Ankit Gupta
Numerade Educator
00:52

Problem 60

Solve for the indicated variable.
$$a^{2}+b^{2}=c^{2} \quad \text { for } b \quad(b>0)$$

AG
Ankit Gupta
Numerade Educator
01:00

Problem 61

Solve for the indicated variable.
$$V=\frac{1}{3} \pi r^{2} h \quad \text { for } r \quad(r>0)$$

AG
Ankit Gupta
Numerade Educator
00:53

Problem 62

Solve for the indicated variable.
$$V=\frac{1}{3} s^{2} h \quad \text { for } s \quad(s>0)$$

AG
Ankit Gupta
Numerade Educator
02:17

Problem 63

A corner shelf is to be made from a triangular piece of plywood, as shown in the diagram. Find the distance $x$ that the shelf will extend along the walls. Assume that the walls are at right angles. Round the answer to a tenth of a foot.
(FIGURE CANNOT COPY)

AG
Ankit Gupta
Numerade Educator
01:59

Problem 64

The volume of a box with a square bottom and a height of 4 in. is given by $V(x)=4 x^{2},$ where $x$ is the length (in inches) of the sides of the bottom of the box.
(FIGURE CANNOT COPY)
a. If the volume of the box is 289 in. $^{3}$, find the dimensions of the box.
b. Are there two possible answers to part (a)? Why or why not?

Narayan Hari
Narayan Hari
Numerade Educator
01:34

Problem 65

A square has an area of 50 in. $^{2}$ What are the lengths of the sides? (Round to one decimal place.)

AG
Ankit Gupta
Numerade Educator
08:31

Problem 66

The amount of money $A$ in an account with an interest rate $r$ compounded annually is given by
$$A=P(1+r)^{t}$$
where $P$ is the initial principal and $t$ is the number of years the money is invested.
a. If a 10,000 dollars investment grows to 11,664 dollars after 2 yr, find the interest rate.
b. If a 6000 dollars investment grows to 7392.60 dollars after 2 yr, find the interest rate.
c. Jamal wants to invest 5000 dollars . He wants the money to grow to at least 6500 dollars in 2 yr to cover the cost of his son's first year at college. What interest rate does Jamal need for his investment to grow to 6500 dollars in 2 yr? Round to the nearest hundredth of a percent.

P Krishnamurthy
P Krishnamurthy
Numerade Educator
07:23

Problem 67

A textbook company has discovered that the profit for selling its books is given by
$$P(x)=-\frac{1}{8} x^{2}+5 x$$
where $x$ is the number of textbooks produced (in thousands) and $P(x)$ is the corresponding profit (in thousands of dollars). The graph of the function is shown at the right.
a. Approximate the number of books required to make a profit of 20,000 dollars . [Hint: Let $P(x)=20 . \text { Then complete the square to solve for } x .$ Round to one decimal place.
b. Why are there two answers to part (a)?

P Krishnamurthy
P Krishnamurthy
Numerade Educator
01:29

Problem 68

If we ignore air resistance, the distance (in feet) that an object travels in free fall can be approximated by $d(t)=16 t^{2},$ where $t$ is the time in seconds after the object was dropped.
a. If the CN Tower in Toronto is 1815 ft high, how long will it take an object to fall from the top of the building? Round to one decimal place.
b. If the Renaissance Tower in Dallas is 886 ft high, how long will it take an object to fall from the top of the building? Round to one decimal place.

Narayan Hari
Narayan Hari
Numerade Educator