Chapter 2

Find two quadratic equations having the given solutions. (There are many correct answers.) $$2+i, 2-i$$

Sketch the graph of the equation by hand. Verify using a graphing utility. $$y=(x-3)^{2}-7$$

Find the domain of \(x\) in the expression. $$\sqrt{2 x^{2}-8}$$

Find two quadratic equations having the given solutions. (There are many correct answers.) $$3+4 i, 3-4 i$$

Sketch the graph of the equation by hand. Verify using a graphing utility. $$y=4-\frac{1}{3} x^{2}$$

Find the domain of \(x\) in the expression. $$\sqrt[4]{3 x^{2}-20 x-7}$$

For each of the following, find the answer algebraically, numerically, and graphically. (a) Find the \(x\) - and \(y\) -intercepts of the graph of \(y=2 x+2\) (b) Verify that the real numbers -1 and 1 are zero(s) of the function \(f(x)=x^{2}-1\) (c) Find the points of intersection of the graphs of \(y=2 x+2\) and \(y=x^{2}-1\)

The floor of a one-story building is 14 feet longer than it is wide. The building has 1632 square feet of floor space. (a) Draw a diagram that gives a visual representation of the floor space. Represent the width as \(w\) and show the length in terms of \(w\). (b) Write a quadratic equation for the area of the floor in terms of \(w\). (c) Find the length and width of the building floor.

Sketch the graph of the equation by hand. Verify using a graphing utility. $$y=-\frac{1}{2}|x+4|-1$$

Find \(a\) and \(b\) in the equation \(x+\sqrt{x-a}=b\) when the solution is \(x=20 .\) (There are many correct answers.)

Find the domain of \(x\) in the expression. $$\sqrt[4]{2 x^{2}+4 x+3}$$

Rationalize the denominator. $$\frac{12}{5 \sqrt{3}}$$

An above-ground swimming pool with a square base is to be constructed such that the surface area of the pool is 561 square feet. The height of the pool is to be 4 feet (see figure). What should the dimensions of the base be? (Hint: The surface area is \(\left.S=x^{2}+4 x h .\right)\)

Sketch the graph of the equation by hand. Verify using a graphing utility. $$y=|x-2|+10$$

Rationalize the denominator. $$\frac{10}{\sqrt{14}-2}$$

An open box is to be made from a square piece of material by cutting four- centimeter squares from each corner and turning up the sides (see figure). The volume of the finished box is to be 576 cubic centimeters. Find the size of the original piece of material.

Rationalize the denominator. $$\frac{3}{8+\sqrt{11}}$$

Operations with Rational Expressions Simplify the expression. $$\frac{8}{3 x}+\frac{3}{2 x}$$

A projectile is fired straight upward from ground level with an initial velocity of 160 feet per second. (a) At what instant will it be back at ground level? (b) When will the height exceed 384 feet?

Rationalize the denominator. $$\frac{14}{3 \sqrt{10}-1}$$

Operations with Rational Expressions Simplify the expression. $$\frac{2}{x^{2}-4}-\frac{1}{x^{2}-3 x+2}$$

A projectile is fired straight upward from ground level with an initial velocity of 128 feet per second. (a) At what instant will it be back at ground level? (b) When will the height be less than 128 feet?

Find the product. $$(x-6)(3 x-5)$$

Operations with Rational Expressions Simplify the expression. $$\frac{2}{z+2}-\left(3-\frac{2}{z}\right)$$

The numbers \(D\) of doctorate degrees (in thousands) awarded to female students from 1991 through 2012 in the United States can be approximated by the model $$D=0.0743 t^{2}+0.628 t+42.61,0 \leq t \leq 22$$ where \(t\) is the year, with \(t=1\) corresponding to \(1991 .\) (Source: U.S. National Center for Education Statistics) (a) Use a graphing utility to graph the model. (b) Use the zoom and trace features to find when the number of degrees was between 50 and 60 thousand. (c) Algebraically verify your results from part (b).

Find the product. $$(3 x+13)(4 x-7)$$

Operations with Rational Expressions Simplify the expression. $$25 y^{2} \div \frac{x y}{5}$$

You want to determine whether there is a relationship between an athlete's weight \(x\) (in pounds) and the athlete's maximum bench-press weight \(y\) (in pounds). Sample data from 12 athletes are shown below. (Spreadsheet at LarsonPrecalculus.com) (165,170),(184,185),(150,200), (210,255),(196,205),(240,295), (202,190),(170,175),(185,195), (190,185),(230,250),(160,150) (a) Use a graphing utility to plot the data. (b) A model for the data is \(y=1.3 x-36\).Use the graphing utility to graph the equation in the same viewing window used in part (a). (c) Use the graph to estimate the values of \(x\) that predict a maximum bench- press weight of at least 200 pounds. (d) Use the graph to write a statement about the accuracy of the model. If you think the graph indicates that an athlete's weight is not a good indicator of the athlete's maximum bench-press weight, list other factors that might influence an individual's maximum bench-press weight.

Find the product. $$(2 x-9)(2 x+9)$$

Solving a Quadratic Equation Find all real solutions of the equation. $$x^{2}-22 x+121=0$$

use the models below, which approximate the numbers of Bed Bath \(\&\) Beyond stores \(B\) and Williams-Sonoma stores \(W\) for the years 2000 through \(2013,\) where \(t\) is the year, with \(t=0\) corresponding to 2000. (Sources: Bed Bath \(\&\) Beyond, Inc.; Williams-Sonoma, Inc.) Bed Bath \& Beyond: $$B=86.5 t+342,0 \leq t \leq 13$$ Williams-Sonoma: $$W=-2.92 t^{2}+52.0 t+381,0 \leq t \leq 13$$. Solve the inequality \(B(t) \geq 900 .\) Explain what the solution of the inequality represents.

The average salaries \(S\) (in thousands of dollars) of secondary classroom teachers in the United States from 2005 through 2013 can be approximated by the model $$S=-0.143 t^{2}+3.73 t+32.5,5 \leq t \leq 13$$ where \(t\) represents the year, with \(t=5\) corresponding to \(2005 .\) (a) Determine algebraically when the average salary of a secondary classroom teacher was \(\$ 50,000\). (b) Verify your answer to part (a) by creating a table of values for the model. (c) Use a graphing utility to graph the model. (d) Use the model to determine when the average salary reached \(\$ 55,500\) (e) Do you believe the model could be used to predict the average salaries for years beyond \(2013 ?\) Explain your reasoning.

Find the product. $$(4 x+1)^{2}$$

Solving a Quadratic Equation Find all real solutions of the equation. $$x(x-20)+3(x-20)=0$$

use the models below, which approximate the numbers of Bed Bath \(\&\) Beyond stores \(B\) and Williams-Sonoma stores \(W\) for the years 2000 through \(2013,\) where \(t\) is the year, with \(t=0\) corresponding to 2000. (Sources: Bed Bath \(\&\) Beyond, Inc.; Williams-Sonoma, Inc.) Bed Bath \& Beyond: $$B=86.5 t+342,0 \leq t \leq 13$$ Williams-Sonoma: $$W=-2.92 t^{2}+52.0 t+381,0 \leq t \leq 13$$. Solve the inequality \(W(t) \leq 600 .\) Explain what the solution of the inequality represents.

The total public debt \(D\) (in trillions of dollars) in the United States from 2005 through 2014 can be approximated by the model $$D=0.051 t^{2}+0.20 t+5.0,5 \leq t \leq 14$$ where \(t\) represents the year, with \(t=5\) corresponding to \(2005 .\) (a) Determine algebraically when the total public debt reached \(\$ 10\) trillion. (b) Verify your answer to part (a) by creating a table of values for the model. (c) Use a graphing utility to graph the model. (d) Use the model to predict when the total public debt will reach \(\$ 20\) trillion. (e) Do you believe the model could be used to predict the total public debt for years beyond \(2014 ?\) Explain your reasoning.

use the models below, which approximate the numbers of Bed Bath \(\&\) Beyond stores \(B\) and Williams-Sonoma stores \(W\) for the years 2000 through \(2013,\) where \(t\) is the year, with \(t=0\) corresponding to 2000. (Sources: Bed Bath \(\&\) Beyond, Inc.; Williams-Sonoma, Inc.) Bed Bath \& Beyond: $$B=86.5 t+342,0 \leq t \leq 13$$ Williams-Sonoma: $$W=-2.92 t^{2}+52.0 t+381,0 \leq t \leq 13$$. Solve the equation \(B(t)=W(t) .\) Explain what the solution of the equation represents.

The metabolic rate of an ectothermic organism increases with increasing temperature within a certain range. Experimental data for oxygen consumption \(C\) (in microliters per gram per hour) of a beetle at certain temperatures yielded the model $$C=0.45 x^{2}-1.73 x+52.65,10 \leq x \leq 25$$ where \(x\) is the air temperature (in degrees Celsius). (a) Use a graphing utility to graph the consumption model over the specified domain. (b) Use the graph to approximate the air temperature resulting in oxygen consumption of 150 microliters per gram per hour. (c) When the temperature is increased from \(10^{\circ} \mathrm{C}\) to \(20^{\circ} \mathrm{C},\) the oxygen consumption will be increased by approximately what factor?

use the models below, which approximate the numbers of Bed Bath \(\&\) Beyond stores \(B\) and Williams-Sonoma stores \(W\) for the years 2000 through \(2013,\) where \(t\) is the year, with \(t=0\) corresponding to 2000. (Sources: Bed Bath \(\&\) Beyond, Inc.; Williams-Sonoma, Inc.) Bed Bath \& Beyond: $$B=86.5 t+342,0 \leq t \leq 13$$ Williams-Sonoma: $$W=-2.92 t^{2}+52.0 t+381,0 \leq t \leq 13$$. Solve the inequality \(B(t) \geq W(t) .\) Explain what the solution of the inequality represents.

The fuel efficiency \(F\) (in miles per gallon) of a car is approximated by $$F=-0.0191 s^{2}+1.639 s+2.20,5 \leq s \leq 65$$ where \(s\) is the average speed of the car (in miles per hour). (a) Use a graphing utility to graph the function over the specified domain. (b) Use the graph to determine the greatest fuel efficiency of the car. How fast should the car travel? (c) When the average speed of the car is increased from 20 miles per hour to 30 miles per hour, the fuel efficiency will be increased by approximately what factor?

Two planes leave simultaneously from Chicago's O'Hare Airport, one flying due north and the other due east (see figure). The northbound plane is flying 50 miles per hour faster than the eastbound plane. After 3 hours, the planes are 2440 miles apart. Find the speed of each plane.

Determine whether the statement is true or false. Justify your answer. The quadratic equation \(-3 x^{2}-x=10\) has two real solutions.

Determine whether the statement is true or false. Justify your answer. If \((2 x-3)(x+5)=8,\) then \(2 x-3=8\) or \(x+5=8\).

Determine whether the statement is true or false. Justify your answer. If \(-10 \leq x \leq 8,\) then \(-10 \geq-x\) and \(-x \geq-8\).

Determine whether the statement is true or false. Justify your answer. The solution set of the inequality \(\frac{3}{2} x^{2}+3 x+6 \geq 0\) is the entire set of real numbers.

Given that \(a\) and \(b\) are nonzero real numbers, determine the solutions of the equations. (a) \(a x^{2}+b x=0\) (b) \(a x^{2}-a x=0\)

Determine whether the statement is true or false. Justify your answer. The domain of \(\sqrt[3]{6-x}\) is \((-\infty, 6]\).

Given that the solutions of a quadratic equation are \(x=(-b \pm \sqrt{b^{2}-4 a c}) /(2 a),\) show that the sum of the solutions is \(S=-b / a\).

The arithmetic mean of \(a\) and \(b\) is given by \((a+b) / 2 .\) Order the statements of the proof to show that if \(a

Given that the solutions of a quadratic equation are \(x=(-b \pm \sqrt{b^{2}-4 a c}) /(2 a),\) show that the product of the solutions is \(P=c / a\).