Chapter 2

Simplify the complex number and write it in standard form. $$4 i^{2}-2 i^{3}$$

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any \(x\) -intercepts of the graph, (c) set \(y=0\) and solve the resulting equation, and (d) compare the result of part (c) with the \(x\) -intercepts of the graph. $$y=\sqrt{11 x-30}-x$$

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$2 x^{3}+5 x^{2}>6 x+9$$

Solve the equation using any convenient method. $$x^{2}-3 x-4=0$$

Determine any point(s) of intersection algebraically. Then verify your result numerically by creating a table of values for each function. $$\begin{aligned} &y=x^{2}-x+1\\\ &y=x^{2}+2 x+4 \end{aligned}$$

The volume of a rectangular package is 2304 cubic inches. The length of the package is 3 times its width, and the height is \(1 \frac{1}{2}\) times its width. (a) Draw a diagram that illustrates the problem. Label the height, width, and length accordingly. (b) Find the dimensions of the package. Use a graphing utility to verify your result.

Simplify the complex number and write it in standard form. $$(\sqrt{-75})^{3}$$

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any \(x\) -intercepts of the graph, (c) set \(y=0\) and solve the resulting equation, and (d) compare the result of part (c) with the \(x\) -intercepts of the graph. $$y=2 x-\sqrt{15-4 x}$$

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$2 x^{3}+3 x^{2}<11 x+6$$

Solve the equation using any convenient method. $$11 x^{2}+33 x=0$$

Determine any point(s) of intersection algebraically. Then verify your result numerically by creating a table of values for each function. $$\begin{aligned} &y=-x^{2}+3 x+1\\\ &y=-x^{2}-2 x-4 \end{aligned}$$

Simplify the complex number and write it in standard form. $$(\sqrt{-2})^{6}$$

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any \(x\) -intercepts of the graph, (c) set \(y=0\) and solve the resulting equation, and (d) compare the result of part (c) with the \(x\) -intercepts of the graph. $$y=3 x-3 \sqrt{x}-4$$

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$x^{3}-3 x^{2}-x>-3$$

Solve the equation using any convenient method. $$(x+3)^{2}=81$$

Use a graphing utility to approximate any points of intersection of the graphs of the equations. Check your results algebraically. $$\begin{aligned} &y=9-2 x\\\ &y=x-3 \end{aligned}$$

Simplify the complex number and write it in standard form. $$\frac{1}{i^{3}}$$

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any \(x\) -intercepts of the graph, (c) set \(y=0\) and solve the resulting equation, and (d) compare the result of part (c) with the \(x\) -intercepts of the graph. $$y=\sqrt{7 x+36}-\sqrt{5 x+16}-2$$

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$2 x^{3}+13 x^{2}-8 x-46 \geq 6$$

Solve the equation using any convenient method. $$(x-1)^{2}=-1$$

Use a graphing utility to approximate any points of intersection of the graphs of the equations. Check your results algebraically. $$\begin{aligned} &x-3 y=-3\\\ &5 x-2 y=11 \end{aligned}$$

The average July 2013 temperature in the contiguous United States was \(74.3^{\circ} \mathrm{F}\). What was the average temperature in degrees Celsius?

Simplify the complex number and write it in standard form. $$\frac{1}{(2 i)^{3}}$$

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any \(x\) -intercepts of the graph, (c) set \(y=0\) and solve the resulting equation, and (d) compare the result of part (c) with the \(x\) -intercepts of the graph. $$y=\frac{1}{x}-\frac{4}{x-1}-1$$

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$3 x^{2}-11 x+16 \leq 0$$

Solve the equation using any convenient method. $$x^{2}-2 x=-\frac{13}{4}$$

Use a graphing utility to approximate any points of intersection of the graphs of the equations. Check your results algebraically. $$\begin{aligned} &y=x\\\ &y=2 x-x^{2} \end{aligned}$$

An executive flew in the corporate jet to a meeting in a city 1500 kilometers away. After traveling the same amount of time on the return flight, the pilot mentioned that they still had 300 kilometers to go. The air speed of the plane was 600 kilometers per hour. How fast was the wind blowing? (Assume that the wind direction was parallel to the flight path and constant all day.)

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any \(x\) -intercepts of the graph, (c) set \(y=0\) and solve the resulting equation, and (d) compare the result of part (c) with the \(x\) -intercepts of the graph. $$y=x-5+\frac{7}{x+3}$$

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$4 x^{2}+12 x+9 \leq 0$$

Solve the equation using any convenient method. $$x^{2}+4 x=-\frac{19}{4}$$

Use a graphing utility to approximate any points of intersection of the graphs of the equations. Check your results algebraically. $$\begin{aligned} &y=4-x^{2}\\\ &y=2 x-1 \end{aligned}$$

A gondola tower in an amusement park casts a shadow that is 80 feet long, while a sign that is 4 feet tall casts a shadow that is \(3 \frac{1}{2}\) feet long. Draw a diagram for the situation. Then find the height of the tower.

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any \(x\) -intercepts of the graph, (c) set \(y=0\) and solve the resulting equation, and (d) compare the result of part (c) with the \(x\) -intercepts of the graph. $$y=x+\frac{9}{x+1}-5$$

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$4 x^{2}-4 x+1>0$$

Solve the equation using any convenient method. $$5 x^{2}=3 x+1$$

Use a graphing utility to approximate any points of intersection of the graphs of the equations. Check your results algebraically. $$\begin{aligned} &x^{3}-y=3\\\ &2 x+y=5 \end{aligned}$$

You have a uniform beam of length \(L\) with a fulcrum \(x\) feet from one end. Objects with weights \(W_{1}\) and \(W_{2}\) are placed at opposite ends of the beam (see figure). The beam will balance when $$W_{1} x=W_{2}(L-x)$$ Find \(x\) such that the beam will balance. Two children weighing 50 pounds \(\left(W_{1}\right)\) and 75 pounds \(\left(W_{2}\right)\) are going to play on a seesaw that is 10 feet long.

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any \(x\) -intercepts of the graph, (c) set \(y=0\) and solve the resulting equation, and (d) compare the result of part (c) with the \(x\) -intercepts of the graph. $$y=2 x+\frac{8}{x-5}-2$$

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$x^{2}+3 x+8>0$$

Solve the equation using any convenient method. $$4 x^{2}=7 x+3$$

Use a graphing utility to approximate any points of intersection of the graphs of the equations. Check your results algebraically. $$\begin{aligned} &y=2 x^{2}\\\ &y=x^{4}-2 x^{2} \end{aligned}$$

You have a uniform beam of length \(L\) with a fulcrum \(x\) feet from one end. Objects with weights \(W_{1}\) and \(W_{2}\) are placed at opposite ends of the beam (see figure). The beam will balance when $$W_{1} x=W_{2}(L-x)$$ Find \(x\) such that the beam will balance. A person weighing 200 pounds \(\left(W_{1}\right)\) is attempting to move a 550 -pound rock \(\left(W_{2}\right)\) with a bar that is 5 feet long.

Determine whether the statement is true or false. Justify your answer. $$i^{44}+i^{150}-i^{74}-i^{109}+i^{61}=-1$$

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any \(x\) -intercepts of the graph, (c) set \(y=0\) and solve the resulting equation, and (d) compare the result of part (c) with the \(x\) -intercepts of the graph. $$y=|x+1|-2$$

Evaluate the expression in two ways. (a) Calculate entirely on your calculator by storing intermediate results and then rounding the final answer to two decimal places. (b) Round both the numerator and denominator to two decimal places before dividing, and then round the final answer to two decimal places. Does the method in part (b) decrease the accuracy? Explain. $$\frac{1+0.73205}{1-0.73205}$$

Determine whether the statement is true or false. Justify your answer. The sum of two imaginary numbers is always an imaginary number.

Graphical Analysis (a) use a graphing utility to graph the equation, (b) use the graph to approximate any \(x\) -intercepts of the graph, (c) set \(y=0\) and solve the resulting equation, and (d) compare the result of part (c) with the \(x\) -intercepts of the graph. $$y=|x-2|-3$$

Evaluate the expression in two ways. (a) Calculate entirely on your calculator by storing intermediate results and then rounding the final answer to two decimal places. (b) Round both the numerator and denominator to two decimal places before dividing, and then round the final answer to two decimal places. Does the method in part (b) decrease the accuracy? Explain. $$\frac{1+0.86603}{1-0.86603}$$

Determine whether the statement is true or false. Justify your answer. The volume of a cube with a side length of 9.5 inches is greater than the volume of a sphere with a radius of 5.9 inches.