Chapter 5

Solve each equation. $$ 5 x^{2}+5=0 $$

Solve each equation by using the Square Root Property. \(x^{2}+x+\frac{1}{4}=\frac{9}{16}\)

NUMBER THEORY Use a quadratic equation to find two real numbers that satisfy each situation, or show that no such numbers exist. Their sum is \(-9\) and their product is 24

Solve \(x^{3}=9 x\) by factoring.

Complete parts a-c for each quadratic function. a. Find the \(y\) -intercept, the equation of the axis of symmetry, and the \(x\) -coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function. $$ f(x)=-0.25 x^{2}-3 x $$

Solve each equation by using the method of your choice. Find exact solutions. \(10 x^{2}+3 x=0\)

For Exercises \(43-45,\) use the following information.The girls' softball team is sponsoring a fund-raising trip to see a professional baseball game. They charter a \(60-\) passenger bus for \(\$ 525 .\) In order to make a profit, they will charge \(\$ 15\) per person if all seats on the bus are sold, but for each empty seat, they will increase the price by \(\$ 1.50\) per person. Write a quadratic function giving the softball team's profit \(P(n)\) from this fund-raiser as a function of the number of passengers \(n .\)

Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. $$ y=-2 x^{2}+20 x-35 $$

Solve each equation. $$ 4 x^{2}+64=0 $$

Solve each equation by using the Square Root Property. \(x^{2}+1.4 x+0.49=0.81\)

NUMBER THEORY Use a quadratic equation to find two real numbers that satisfy each situation, or show that no such numbers exist. Their sum is 12 and their product is \(-28\)

Complete parts a-c for each quadratic function. a. Find the \(y\) -intercept, the equation of the axis of symmetry, and the \(x\) -coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function. $$ f(x)=\frac{1}{2} x^{2}+3 x+\frac{9}{2} $$

Solve each equation by using the method of your choice. Find exact solutions. \(2 x^{2}-12 x+7=5\)

For Exercises \(43-45,\) use the following information.The girls' softball team is sponsoring a fund-raising trip to see a professional baseball game. They charter a \(60-\) passenger bus for \(\$ 525 .\) In order to make a profit, they will charge \(\$ 15\) per person if all seats on the bus are sold, but for each empty seat, they will increase the price by \(\$ 1.50\) per person. What is the minimum number of passengers needed in order for the softball team not to lose money?

Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. $$ y=3 x^{2}+3 x-1 $$

Solve each equation. $$ 2 x^{2}+12=0 $$

Solve each equation by using the Square Root Property. \(4 x^{2}-28 x+49=5\)

LAW ENFORCEMENT Police officers can use the length of skid marks to help determine the speed of a vehicle before the brakes were applied. If the skid marks are on dry concrete, the formula \(\frac{s^{2}}{24}=d\) can be used. In the formula, s represents the speed in miles per hour and \(d\) represents the length of the skid marks in feet. If the length of the skid marks on dry concrete are 50 feet, how fast was the car traveling?

Complete parts a-c for each quadratic function. a. Find the \(y\) -intercept, the equation of the axis of symmetry, and the \(x\) -coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function. $$ f(x)=x^{2}-\frac{2}{3} x-\frac{8}{9} $$

Solve each equation by using the method of your choice. Find exact solutions. \(21=(x-2)^{2}+5\)

For Exercises \(43-45,\) use the following information.The girls' softball team is sponsoring a fund-raising trip to see a professional baseball game. They charter a \(60-\) passenger bus for \(\$ 525 .\) In order to make a profit, they will charge \(\$ 15\) per person if all seats on the bus are sold, but for each empty seat, they will increase the price by \(\$ 1.50\) per person. What is the maximum profit the team can make with this fund-raiser, and how many passengers will it take to achieve this maximum?

Write each quadratic function in vertex form, if not already in that form. Then identify the vertex, axis of symmetry, and direction of opening. $$ y=4 x^{2}-12 x-11 $$

Solve each equation. $$ 6 x^{2}+72=0 $$

Solve each equation by using the Square Root Property. \(9 x^{2}+30 x+25=11\)

Write a quadratic equation with the given graph or roots. \(-\frac{2}{3}, \frac{3}{4}\)

Determine whether each function has a maximum or a minimum value and find the maximum or minimum value. Then state the domain and range of the function. $$ f(x)=2 x+2 x^{2}+5 $$

The supporting cables of the Golden Gate Bridge approximate the shape of a parabola. The parabola can be modeled by \(y=0.00012 x^{2}+6,\) where \(x\) represents the distance from the axis of symmetry and \(y\) represents the height of the cables. The related quadratic equation is \(0.00012 x^{2}+6=0\). Calculate the value of the discriminant.

REASONING Examine the graph of \(y=x^{2}-4 x-5\) a. What are the solutions of \(0=x^{2}-4 x-5 ?\) b. What are the solutions of \(x^{2}-4 x-5 \geq 0 ?\) c. What are the solutions of \(x^{2}-4 x-5 \leq 0 ?\)

Write an equation for a parabola with vertex at the origin and that passes through \((2,-8) .\)

Find the values of \(m\) and \(n\) that make each equation true. $$ 8+15 i=2 m+3 n i $$

Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. \(x^{2}+0.6 x+c\)

OPEN ENDED Give an example of a quadratic equation with a double root, and state the relationship between the double root and the graph of the related function.

Write a quadratic equation with the given graph or roots. \(-\frac{3}{2},-\frac{4}{5}\)

Determine whether each function has a maximum or a minimum value and find the maximum or minimum value. Then state the domain and range of the function. $$ f(x)=x-2 x^{2}-1 $$

The supporting cables of the Golden Gate Bridge approximate the shape of a parabola. The parabola can be modeled by \(y=0.00012 x^{2}+6,\) where \(x\) represents the distance from the axis of symmetry and \(y\) represents the height of the cables. The related quadratic equation is \(0.00012 x^{2}+6=0\). What does the discriminant tell you about the supporting cables of the Golden Gate Bridge?

OPEN ENDED List three points you might test to find the solution of \((x+3)(x-5)<0\)

Write an equation for a parabola with vertex at \((-3,-4)\) and \(y\) -intercept 8

Find the values of \(m\) and \(n\) that make each equation true. $$ (m+1)+3 n i=5-9 i $$

Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. \(x^{2}-2.4 x+c\)

REASONING Explain how you can estimate the solutions of a quadratic equation by examining the graph of its related function.

To avoid hitting any rocks below, a cliff diver jumps up and out. The equation \(h=-16 t^{2}\) \(+4 t+26\) describes her height \(h\) in feet \(t\) seconds after jumping. Find the time at which she returns to a height of 26 feet.

Determine whether each function has a maximum or a minimum value and find the maximum or minimum value. Then state the domain and range of the function. $$ f(x)=-7-3 x^{2}+12 x $$

Civil engineers are designing a section of road that is going to dip below sea level. The road’s curve can be modeled by the equation \(y=0.00005 x^{2}-0.06 x,\) where \(x\) is the horizontal distance in feet between the points where the road is at sea level and \(y\) is the elevation (a positive value being above sea level and a negative being below). The engineers want to put stop signs at the locations where the elevation of the road is equal to sea level. At what horizontal distances will they place the stop signs?

CHALLENGE Graph the intersection of the graphs of \(y \leq-x^{2}+4\) and \(y \geq x^{2}-4\)

Write one sentence that compares the graphs of \(y=0.2(x+3)^{2}+1\) and \(y=0.4(x+3)^{2}+1\)

Find the values of \(m\) and \(n\) that make each equation true. $$ (2 m+5)+(1-n) i=-2+4 i $$

Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. \(x^{2}-\frac{8}{3} x+c\)

CHALLENGE A quadratic function has values \(f(-4)=-11, f(-2)=9\) , and \(f(0)=5 .\) Between which two \(x\) -values must \(f(x)\) have a zero? Explain your reasoning.

Lumber companies need to be able to estimate the number of board feet that a given log will yield. One of the most commonly used formulas for estimating board feet is the Doyle Log Rule, \(B=\frac{L}{16}\left(D^{2}-\right.\) \(8 D+16 )\) where \(B\) is the number of board feet, \(D\) is the diameter in inches, and \(L\) is the length of the log in feet. Rewrite Doyle’s formula for logs that are 16 feet long.

Determine whether each function has a maximum or a minimum value and find the maximum or minimum value. Then state the domain and range of the function. $$ f(x)=-20 x+5 x^{2}+9 $$