Chapter 3

Solve each equation. For equations with real solutions, support your answers graphically. $$x(12 x+11)=-2$$

For quadratic function, (a) use the vertex formula to find the coordinates of the vertex and (b) graph the function. Do not use a calculator. $$P(x)=2 x^{2}-8 x+9$$

Solve each equation. For equations with real solutions, support your answers graphically. $$-4+9 x-2 x^{2}=0$$

Solve each problem. A frame for a picture is 2 inches wide. The picture inside the frame is 4 inches longer than it is wide. See the figure at the top of the next column. If the area of the picture is 320 square inches, find the outside dimensions of the picture frame.

For quadratic function, (a) use the vertex formula to find the coordinates of the vertex and (b) graph the function. Do not use a calculator. $$P(x)=-3 x^{2}+24 x-46$$

Solve each problem. Suppose the revenue \(R\) in thousands of dollars that a company receives from producing \(x\) thousand tablets is \(R(x)=x(40-2 x)\). (a) Evaluate \(R(2)\) and interpret the result. (b) How many tablets should the company produce to maximize its revenue? (c) What is the maximum revenue?

Determine whether each statement is true or false. If it is false, tell why. $$i \sqrt{-16}$$

For quadratic function, (a) use the vertex formula to find the coordinates of the vertex and (b) graph the function. Do not use a calculator. $$P(x)=-2 x^{2}-6 x-5$$

Solve each equation. For equations with real solutions, support your answers graphically. $$\frac{1}{3} x^{2}-\frac{1}{3} x=24$$

Solve each problem. The manager of an 80-unit apartment complex knows from experience that at a rent of \(\$ 400\) per month, all units will be rented. However, for each increase of \(\$ 20\) in rent, he can expect one unit to be vacated. Let \(x\) represent the number of \(\$ 20\) increases over \(\$ 400\). (a) Express, in terms of \(x\), the number of apartments that will be rented if \(x\) increases of \(\$ 20\) are made. (For example, with three such increases, the number of apartments rented will be \(80-3=77\).) (b) Express the rent per apartment if \(x\) increases of \(\$ 20\) are made. (For example, if he increases rent by \(\mathrm{S} 60=3 \times \$ 20,\) the rent per apartment is given by \(400+3(20)=\$ 460 .)\) (c) Determine a revenue function \(R\) in terms of \(x\) that will give the revenue generated as a function of the number of \(\$ 20\) increases. (d) For what number of increases will the revenue be \(\$ 37,500 ?\) (e) What rent should he charge in order to achieve the maximum revenue?

Multiply or divide as indicated. Simplify each answer. $$\sqrt{-13} \cdot \sqrt{-13}$$

For quadratic function, (a) use the vertex formula to find the coordinates of the vertex and (b) graph the function. Do not use a calculator. $$P(x)=2 x^{2}-2 x+1$$

Solve each equation. For equations with real solutions, support your answers graphically. $$\frac{1}{6} x^{2}+\frac{1}{6} x=5$$

Solve each problem. When Respect Brings Success charges \(\$ 600\) for a seminar on management techniques, it attracts 1000 people. For each decrease of \(\$ 20\) in the charge, an additional 100 people will attend the seminar. Let \(x\) represent the number of \(\$ 20\) decreases in the charge. (a) Determine a revenue function \(R\) that will give reve enue generated as a function of the number of \(\$ 20\) decreases. (b) Find the value of \(x\) that maximizes the revenue. What should the company charge to maximize the revenue? (c) What is the maximum revenue the company can generate?

Multiply or divide as indicated. Simplify each answer. $$\sqrt{-17} \cdot \sqrt{-17}$$

For \(f(x)=d x^{2}-\frac{1}{2} d x+k, d \neq 0,\) find the \(x\) -coordinate of the vertex.

Solve each equation. For equations with real solutions, support your answers graphically. $$(x+2)(x-1)=7 x+5$$

Multiply or divide as indicated. Simplify each answer. $$\sqrt{-3} \cdot \sqrt{-8}$$

Graph each function in a viewing window that will allow you to use your calculator to approximate (a) the coordinates of the vertex and (b) the \(x\) -intercepts. Give values to the nearest hundredth. $$P(x)=-0.32 x^{2}+\sqrt{3} x+2.86$$

Solve each equation. For equations with real solutions, support your answers graphically. $$(x+4)(x-1)=-5 x-4$$

Multiply or divide as indicated. Simplify each answer. $$\sqrt{-5} \cdot \sqrt{-15}$$

Graph each function in a viewing window that will allow you to use your calculator to approximate (a) the coordinates of the vertex and (b) the \(x\) -intercepts. Give values to the nearest hundredth. $$P(x)=-\sqrt{2} x^{2}+0.45 x+1.39$$

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}-2 x-4=0$$

Solve each problem. A frog leaps from a stump 3 feet high and lands 4 feet from the base of the stump. We can consider the initial position of the frog to be at \((0,3)\) and its landing position to be at \((4,0)\). It is determined that the height \(h\) in feet of the frog as a function of its distance \(x\) from the base of the stump is given by $$h(x)=-0.5 x^{2}+1.25 x+3$$ (a) How high was the frog when its horizontal distance \(x\) from the base of the stump was 2 feet? (b) What was the horizontal distance from the base of the stump when the frog was 3.25 feet above the ground? (c) At what horizontal distance from the base of the stump did the frog reach its highest point? (d) What was the maximum height reached by the frog?

Graph each function in a viewing window that will allow you to use your calculator to approximate (a) the coordinates of the vertex and (b) the \(x\) -intercepts. Give values to the nearest hundredth. $$y=1.34 x^{2}-3 x+\sqrt{5}$$

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}+8 x+13=0$$

Solve each problem. Refer to Exercise \(37 .\) Suppose that the initial position of the frog is \((0,4)\) and its landing position is \((6,0) .\) The height of the frog in feet is given by $$h(x)=-\frac{1}{3} x^{2}+\frac{4}{3} x+4$$ (a) What was the horizontal distance \(x\) from the base of the stump when the frog reached maximum height? (b) What was the maximum height?

Graph each function in a viewing window that will allow you to use your calculator to approximate (a) the coordinates of the vertex and (b) the \(x\) -intercepts. Give values to the nearest hundredth. $$y=-0.55 x^{2}+3.21 x$$

Solve each equation. For equations with real solutions, support your answers graphically. $$2 x^{2}+2 x=-1$$

Solve each problem. To determine the appropriate landing speed of an airplane, we might use $$f(x)=\frac{1}{10} x^{2}-3 x+22$$ where \(x\) is the initial landing speed in feet per second and \(f(x)\) is the length of the runway in feet. If the landing speed is too fast, the plane may run out of runway; if the speed is too slow, the plane may stall. If the runway is 800 feet long, what is the appropriate landing speed? What is the landing speed in mph? (Hint: 5280 feet = 1 mile)

Graph each function in a viewing window that will allow you to use your calculator to approximate (a) the coordinates of the vertex and (b) the \(x\) -intercepts. Give values to the nearest hundredth. $$y=-1.24 x^{2}+1.68 x$$

Solve each equation. For equations with real solutions, support your answers graphically. $$9 x^{2}-12 x=-8$$

Solve each problem. As a function of age group \(x,\) the fatality rate (per \(100,000\) population) for males killed in automobile accidents can be approximated by $$f(x)=1.8 x^{2}-12 x+37.4$$ where \(x=0\) represents ages \(21-24, x=1\) represents ages \(25-34, x=2\) represents ages \(35-44,\) and so on. Find the age group at which the accident rate is a minimum, and find the minimum rate. (Source: National Highway Traffic Safety Administration.)

Solve each equation. For equations with real solutions, support your answers graphically. $$x(x-1)=1$$

Solve each problem. The table lists the average heating bill for a natural gas consumer in Indiana during various months of the year. $$\begin{array}{|c|c|} \hline \text { Month } & \text { Bill ( } \$ \text { ) } \\ \hline \text { Jan. } & 108 \\ \text { Mar. } & 68 \\ \text { May } & 18 \\ \text { July } & 12 \\ \text { Sept. } & 13 \\ \text { Nov. } & 54 \\\\\hline \end{array}$$ (a) Plot the data. Let \(x=1\) correspond to January, \(x=2\) to February, and so on. (b) Find a quadratic function \(f(x)=a(x-h)^{2}+k\) that models the data. Use \((7,12)\) as the vertex and \((1,108)\) as another point to determine \(a\) (c) Plot the data together with the graph of \(f\) in the same window. How well does \(f\) model the average heating bill over these months? (d) Use the quadratic regression feature of a graphing calculator to determine the quadratic function \(g\) that provides the best fit for the data. (e) Use the functions \(f\) and \(g\) to approximate the heating bill to the nearest dollar in the following months. (i) February (ii) June

Multiply or divide as indicated. Simplify each answer. $$\frac{\sqrt{-10}}{\sqrt{-40}}$$

Solve each equation. For equations with real solutions, support your answers graphically. $$x(x-3)=2$$

Solve each problem. The table lists the projected number of shipments \(S\) of a product, in millions, \(x\) years after 2017 . $$\begin{array}{|c|c|} \hline \text { Year } & S \\ \hline 0 & 23 \\ 1 & 15 \\ 2 & 11 \\ 3 & 8 \\ 4 & 7 \\ \hline \end{array}$$ (a) Evaluate \(S(3)\) and interpret the result. (b) Use quadratic regression to find a function \(f\) that models these data.

Multiply or divide as indicated. Simplify each answer. $$\frac{\sqrt{-15}}{\sqrt{-135}}$$

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}-5 x=x-7$$

Solve each problem. Selected values of the stopping distance \(y\) in feet of a car traveling \(x\) mph are given in the table. $$\begin{array}{|c|c|}\hline \begin{array}{c} \text { Speed } \\ \text { (in mph) } \end{array} & \begin{array}{c} \text { Stopping Distance } \\ \text { (in feet) } \end{array} \\ \hline 20 & 46 \\ 30 & 87 \\ 40 & 140 \\ 50 & 240 \\ 60 & 282 \\ 70 & 371 \\ \hline \end{array}$$ (a) Plot the data. (b) The quadratic function $$f(x)=0.056057 x^{2}+1.06657 x$$ is one model for the data. Find and interpret \(f(45)\) (c) Graph the function in the same window as the data to determine how well \(f\) models the stopping distance.

Solve each equation. For equations with real solutions, support your answers graphically. $$11 x^{2}-3 x+2=4 x+1$$

Solve each problem. The coast-down time \(y\) for a typical car as it drops \(10 \mathrm{mph}\) from an initial speed \(x\) depends on several factors, such as average drag, tire pressure, and whether the transmission is in neutral. The table gives the coast-down time in seconds for a car under standard conditions for selected speeds in miles per hour. $$\begin{array}{|c|c|} \hline \begin{array}{c} \text { Initial Speed } \\ \text { (in mph) } \end{array} & \begin{array}{c} \text { Coast-Down } \\ \text { Time (in seconds) } \end{array} \\ \hline 30 & 30 \\ 35 & 27 \\ 40 & 23 \\ 45 & 21 \\ 50 & 18 \\ 55 & 16 \\ 60 & 15 \\ 65 & 13 \\ \hline \end{array}$$ (a) Plot the data. (b) Use the quadratic regression feature of a graphing calculator to find the quadratic function \(g\) that best fits the data. Graph this function in the same window as the data. Is \(g\) a good model for the data? (c) Use \(g\) to predict the coast-down time, to the nearest second, at an initial speed of 70 mph. (d) Use the graph to find the speed that comesponds to a coast-down time of 24 seconds.

Multiply or divide as indicated. Simplify each answer. $$\frac{\sqrt{-12} \cdot \sqrt{-6}}{\sqrt{8}}$$

Solve each equation. For equations with real solutions, support your answers graphically. $$4 x^{2}-12 x=-11$$

Solve each problem. Suppose that a person's heart rate, \(x\) minutes after vigorous exercise has stopped, can be modeled by $$f(x)=\frac{4}{5}(x-10)^{2}+80$$ The output is in beats per minute, where the domain of \(f(x)\) is \(0 \leq x \leq 10\) (a) Evaluate \(f(0)\) and \(f(2) .\) Interpret the result. (b) Estimate the times when the person's heart rate was between 100 and 120 beats per minute, inclusive.

Add or subtract as indicated. Write each sum or difference in standard form. $$(3+2 i)+(4-3 i)$$

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}=2 x-5$$

Solve each problem. An athlete's heart rate \(R\) in beats per minute after \(x\) minutes is given by $$R(x)=2(x-4)^{2}+90.$$ where \(0 \leq x \leq 8\) (a) Describe the heart rate during this period of time. (b) Determine the minimum heart rate during this 8 -minute period.

Solve each equation. For equations with real solutions, support your answers graphically. $$\frac{1}{3} x^{2}+\frac{1}{4} x-3=0$$