Chapter 8

Find the exact solution \((s)\) to each problem. If the solution(s) are irrational, then also find approximate solution(s) to the nearest tenth. Tossing a ball. A ball is tossed into the air at \(10 \mathrm{ft} / \mathrm{sec}\) from a height of 5 feet. How long does it take to reach the earth?

Find all real or imaginary solutions to each equation. Use the method of your choice. $$m^{2}+2 m-24=0$$

Flying high. An arrow is shot straight upward with a velocity of 96 feet per second (ft/sec) from an altitude of 6 feet. For how many seconds is this arrow more than 86 feet high?Putting the shot. In 1978 Udo Beyer (East Germany) set a world record in the shot-put of \(72 \mathrm{ft} 8\) in. If Beyer had projected the shot straight upward with a velocity of \(30 \mathrm{ft} / \mathrm{sec}\) from a height of \(5 \mathrm{ft}\), then for what values of \(t\) would the shot be under 15 ft high?

Find all real or imaginary solutions to each equation. Use the method of your choice. $$q^{2}+6 q-7=0$$

Find the exact solution \((s)\) to each problem. If the solution(s) are irrational, then also find approximate solution(s) to the nearest tenth. Penny tossing. If a penny is thrown downward at \(30 \mathrm{ft} / \mathrm{sec}\) from the bridge at Royal Gorge, Colorado, how long does it take to reach the Arkansas River 1000 ft below?

Find all real or imaginary solutions to each equation. Use the method of your choice. $$(x-2)^{2}=-9$$

Find all real or imaginary solutions to each equation. Use the method of your choice. $$(2 x-1)^{2}=-4$$

Find all real or imaginary solutions to each equation. Use the method of your choice. $$-x^{2}+x+6=0$$

Solve each problem. Recovering an investment. The manager at Cream of the Crop bought a load of watermelons for \(\$ 200 .\) She priced the melons so that she would make \(\$ 1.50\) profit on each melon. When all but 30 had been sold, the manager had recovered her initial investment. How many did she buy originally?

Work in a small group to solve \(a x^{2}+b x+c>0\) for \(x\) in each case. a) \(b^{2}-4 a c=0\) and \(a>0\) b) \(b^{2}-4 a c=0\) and \(a<0\) c) \(b^{2}-4 a c<0\) and \(a>0\) d) \(b^{2}-4 a c<0\) and \(a<0\) e) \(b^{2}-4 a c>0\) and \(a>0\) f) \(b^{2}-4 a c>0\) and \(a<0\)

Find all real or imaginary solutions to each equation. Use the method of your choice. $$-x^{2}+x+12=0$$

Find all real or imaginary solutions to each equation. Use the method of your choice. $$x^{2}-6 x+10=0$$

Solve each problem. Sharing cost. The members of a flying club plan to share equally the cost of a \(\$ 200,000\) airplane. The members want to find five more people to join the club so that the cost per person will decrease by \(\$ 2000 .\) How many members are currently in the club?

Find all real or imaginary solutions to each equation. Use the method of your choice. $$x^{2}-8 x+17=0$$

Find all real or imaginary solutions to each equation. Use the method of your choice. $$2 x-5=\sqrt{7 x+7}$$

Exploration. a) Given that \(P(x)=x^{4}+6 x^{2}-27,\) find \(P(3 i)\) 8 \(P(-3 i), P(\sqrt{3}),\) and \(P(-\sqrt{3})\) b) What can you conclude about the values \(3 i,-3 i, \sqrt{3}\) and \(-\sqrt{3}\) and their relationship to each other?

Solve each problem. Traveling club. The members of a traveling club plan to share equally the cost of a \(\$ 150,000\) motorhome. If they can find 10 more people to join and share the cost, then the cost per person will decrease by \(\$ 1250 .\) How many members are there originally in the club?

Find all real or imaginary solutions to each equation. Use the method of your choice. $$\sqrt{7 x+29}=x+3$$

Cooperative learning. Work with a group to write a quadratic equation that has each given pair of solutions. a) \(3+\sqrt{5}, 3-\sqrt{5}\) b) \(4-2 i, 4+2 i\) c) \(\frac{1+i \sqrt{3}}{2}, \frac{1-i \sqrt{3}}{2}\)

Find the solutions to \(6 x^{2}+5 x-4=0 .\) Is the sum of your solutions equal to \(-\frac{b}{a}\) ? Explain why the sum of the solutions to any quadratic equation is \(-\frac{b}{a}\) (Hint: Use the quadratic formula.)

Find all real or imaginary solutions to each equation. Use the method of your choice. $$\frac{1}{x}+\frac{1}{x-1}=\frac{1}{4}$$

Find all real or imaginary solutions to each equation. Use the method of your choice. $$\frac{1}{x}-\frac{2}{1-x}=\frac{1}{2}$$

Solve each equation by locating the x-intercepts on a calculator graph. Round approximate answers to two decimal places. $$x^{4}-116 x^{2}+1600=0$$

What is the product of the two solutions to \(6 x^{2}+5 x-4=0 ?\) Explain why the product of the solutions to any quadratic equation is \(\frac{c}{a}\)

Solve each equation by locating the x-intercepts on a calculator graph. Round approximate answers to two decimal places. $$\left(x^{2}+3 x\right)^{2}-7\left(x^{2}+3 x\right)+9=0$$

Solve each equation by locating the x-intercepts on a calculator graph. Round approximate answers to two decimal places. $$x^{2}-3 x^{1 / 2}-12=0$$

Solve each problem. The formula \(1211.1 L=C A^{2} S\) is used to determine the approach speed for landing an aircraft, where \(L\) is the gross weight of the aircraft in pounds, \(C\) is the coefficient of lift, \(S\) is the surface area of the wings in square feet ( \(\mathrm{ft}^{2}\) ), and \(A\) is approach speed in feet per second. Find \(A\) for the Piper Cheyenne, which has a gross weight of 8700 lb, a coefficient of lift of \(2.81,\) and wing surface area of \(200 \mathrm{ft}^{2}\).

Determine the number of real solutions to each equation by examining the calculator graph of \(y=a x^{2}+b x+c .\) Use the discriminant to check your conclusions. $$-2 x^{2}-403=0$$

Solve each problem. The period \(T\) (time in seconds for one complete cycle) of a simple pendulum is related to the length \(L\) (in feet) of the pendulum by the formula \(8 T^{2}=\pi^{2} L .\) If a child is on a swing with a 10 -foot chain, then how long does it take to complete one cycle of the swing?

Solve each problem. Tropical Pools figures that its monthly revenue in dollars on the sale of \(x\) above ground pools is given by \(R=1500 x-3 x^{2},\) where \(x\) is less than \(25 .\) What number of pools sold would provide a revenue of \(\$ 17,568 ?\)

Which of the following equations is not a quadratic equation? Explain your answer. a) \(\pi x^{2}-\sqrt{5} x-1=0\) b) \(3 x^{2}-1=0\) c) \(4 x+5=0\) d) \(0.009 x^{2}=0\)

Solve \(x^{2}-4 x+k=0\) for \(k=0,4,5,\) and 10. a) When does the equation have only one solution? b) For what values of \(k\) are the solutions real? c) For what values of \(k\) are the solutions imaginary?

Write a quadratic equation of each of the following types, then trade your equations with those of a classmate. Solve the equations and verify that they are of the required types. a) a single rational solution b) two rational solutions c) two irrational solutions d) two imaginary solutions

For each equation, find approximate solutions rounded to two decimal places. $$x^{2}-7.3 x+12.5=0$$

For each equation, find approximate solutions rounded to two decimal places. $$1.2 x^{2}-\pi x+\sqrt{2}=0$$

For each equation, find approximate solutions rounded to two decimal places. $$2 x-3=\sqrt{20-x}$$

For each equation, find approximate solutions rounded to two decimal places. $$x^{2}-1.3 x=22.3-x^{2}$$