Chapter 12

Use a graphing calculator or a computer to show that the radical equation \(\sqrt{11 x-30}=-x\) has no solution. Explain.

$$\text { Simplify } \frac{3}{5-\sqrt{5}}$$ \(\begin{array}{llll}\text { (A) } \frac{15+3 \sqrt{5}}{20} & \text { (B) } \frac{15+\sqrt{5}}{20} & \text { (C) } \frac{15+\sqrt{15}}{20}\end{array}\) (D) \(\frac{15-3 \sqrt{5}}{20}\)

Simplify the radical expression. $$\sqrt{24}$$

Factor the expression. $$16 x^{2}-25$$

Solve the quadratic equation. $$7 z^{2}-46 z=21$$

Factor the expression completely. \(x^{4}-3 x^{2}-25 x^{2}+75\)

Simplify the radical expression. $$\sqrt{60}$$

Factor the expression. $$x^{2}+18 x+81$$

Solve the quadratic equation. $$11 x^{2}-22=0$$

Does the equation model direct variation, inverse variation, or neither? \(x=\frac{7}{y}\)

Use the following information.Find a partner to help you with the following experiment. Take a ruler and drop it from a fixed height. Once the ruler is released, your partner should catch it as quickly as possible. Measure how far the ruler falls before your partner catches it. The reaction time can be calculated by the equation $$ t=\sqrt{\frac{d}{192}} $$ where \(t\) is the reaction time (in seconds) and \(d\) is the distance (in inches) the ruler falls. Suppose for three trials, the ruler falls 4 inches, 1 inch, and 3 inches. Calculate the average reaction time for the three trials.

Simplify the radical expression. $$\sqrt{175}$$

Factor the expression. $$7 x^{2}-28 x+28$$

Solve the quadratic equation. $$x^{2}+20 x+10=0$$

Does the equation model direct variation, inverse variation, or neither? \(y=8 x\)

Write a radical equation that has a solution of 18 .

Use the following information.Find a partner to help you with the following experiment. Take a ruler and drop it from a fixed height. Once the ruler is released, your partner should catch it as quickly as possible. Measure how far the ruler falls before your partner catches it. The reaction time can be calculated by the equation $$ t=\sqrt{\frac{d}{192}} $$ where \(t\) is the reaction time (in seconds) and \(d\) is the distance (in inches) the ruler falls. Perform the experiment several times, recording the distance the ruler falls each time. Organize the results in a table. Calculate the average reaction time for your first five trials. Compare results with your classmates.

Simplify the radical expression. $$\sqrt{9900}$$

Factor the expression. $$45 x^{2}-60 x+20$$

Make a sketch and write a quadratic equation to model the situation. Then solve the equation. In art class you are designing the floor plan of a house. The kitchen is supposed to have 150 square feet of space. What should the dimensions of the kitchen be if you want it to be square?

Use the following information. A ride at an amusement park spins in a circle of radius \(r\) (in meters). The centripetal force \(F\) experienced by a passenger on the ride is modeled by the equation below, where \(t\) is the number of seconds the ride takes to complete one revolution and \(m\) is the mass (in kilograms) of the passenger. $$t=\sqrt{\frac{4 \pi^{2} m r}{F}}$$ A person whose mass is 67.5 kilograms is on a ride that is spinning in a circle at a rate of 10 seconds per revolution. The radius of the circle is 6 meters. How much centripetal force does the person experience?

Does the equation model direct variation, inverse variation, or neither? \(y=9 x+1\)

Solve the percent problem. What is \(30 \%\) of \(160 ?\)

Simplify the radical expression. $$\sqrt{\frac{20}{25}}$$

Factor the expression. $$-48 x^{2}+216 x-243$$

Make a sketch and write a quadratic equation to model the situation. Then solve the equation. A rectangle is \(2 x\) feet long and \(x+5\) feet wide. The area is 600 square feet. What are the dimensions of the rectangle?

Divide. \((6 z+10) \div 2\)

What is \(30 \%\) of \(160 ?\) 105 is what percent of \(240 ?\)

Simplify the radical expression. $$\frac{1}{2} \sqrt{80}$$

The geometric mean of 16 and a is 32. What is the value of a?

Make a sketch and write a quadratic equation to model the situation. Then solve the equation. The height of a triangle is 4 more than twice its base. The area of the triangle is 60 square centimeters. What are the dimensions of the triangle?

Use the following information. Blotting paper is a thick, soft paper used for absorbing fluids such as water or ink. The distance \(d\) (in centimeters) that tap water is absorbed up a strip of blotting paper at a temperature of \(28.4^{\circ} \mathrm{C}\) is given by the equation \(d=0.444 \sqrt{t}\) where \(t\) is the time (in seconds). Approximately how many minutes would it take for the water to travel a distance of 28 centimeters up the strip of blotting paper?

Divide. \(\left(7 x^{3}-2 x^{2}\right) \div 14 x\)

Sketch the graph of the function. $$y=\frac{1}{x-6}-1$$$$y=\frac{1}{x-6}-1$$

Simplify the radical expression. $$\frac{3 \sqrt{7}}{\sqrt{9}}$$

Postage Stamps In \(1960,\) a first-class United States postage stamp cost \(\$ .04 .\) In \(1999,\) a first-class United States postage stamp cost \(\$ .33 .\) Write a compound inequality that represents the different prices that a postage stamp could have cost between 1960 and \(1999 .\) (Review 6.3 )

The path of a diver diving from a 10 -foot high diving board is $$h=-0.44 x^{2}+2.61 x+10$$ where \(h\) is the height of the diver above water (in feet) and \(x\) is the horizontal distance (in feet) from the end of the board. How far from the end of the board will the diver enter the water?

Use the following information. Blotting paper is a thick, soft paper used for absorbing fluids such as water or ink. The distance \(d\) (in centimeters) that tap water is absorbed up a strip of blotting paper at a temperature of \(28.4^{\circ} \mathrm{C}\) is given by the equation \(d=0.444 \sqrt{t}\) where \(t\) is the time (in seconds). How far up the blotting paper would the water be after \(33 \frac{1}{3}\) seconds?

Sketch the graph of the function. $$y=\frac{1}{x-5}+2$$

Simplify the radical expression. $$4 \sqrt{\frac{11}{16}}$$

You are on a research boat in the ocean. You see a penguin jump out of the water. The path followed by the penguin is given by $$ h=-0.05 x^{2}+1.178 x $$ where \(h\) is the height (in feet) the penguin jumps out of the water and \(x\) is the horizontal distance (in feet) traveled by the penguin over the water. Sketch a graph of the equation.

Use the proportion \(\frac{a}{b}=\frac{b}{d},\) where \(a\) \(\boldsymbol{b},\) and \(\boldsymbol{d}\) are positive numbers. In the proportion \(\frac{a}{b}=\frac{b}{d}, b\) is called the geometric mean of \(a\) and \(d .\) Use the cross product property to show that \(b=\sqrt{a d}\).

Sketch the graph of the function. $$y=\frac{2}{x-6}+9$$

Solve the equation. $$x^{2}+4 x-8=0$$

You are on a research boat in the ocean. You see a penguin jump out of the water. The path followed by the penguin is given by $$ h=-0.05 x^{2}+1.178 x $$ where \(h\) is the height (in feet) the penguin jumps out of the water and \(x\) is the horizontal distance (in feet) traveled by the penguin over the water. How many horizontal feet did the penguin travel over the water before reaching its maximum height?

Use the proportion \(\frac{a}{b}=\frac{b}{d},\) where \(a\) \(\boldsymbol{b},\) and \(\boldsymbol{d}\) are positive numbers. Two numbers have a geometric mean of 4. One number is 6 more than the other. a. Use the proportion in Exercise \(73 .\) Rewrite the proportion substituting the given value for the geometric mean. b. Let \(x\) represent one of the numbers. How can you represent "one number is 6 more than the other" in the proportion? Rewrite the proportion using \(x\) c. Solve the proportion in part (b) to find the numbers.

Find the domain and the range of the function. $$f(x)=\sqrt{x}-3$$

Solve the equation. $$x^{2}-2 x-4=0$$

MULTIPLE CHOICE Which of the following is a solution of the equation \(2 x^{2}+8 x-25=5 ?\) \begin{array}{ccccc}\mathbf{A} & \sqrt{17}+1 & \mathbf{B} & -\sqrt{19}-2 & \mathbf{C} & \sqrt{17}-2 & \mathbf{D} &\sqrt{21} & -2\end{array}

Two numbers have a geometric mean of \(10 .\) One number is 21 less than the other. Find the numbers.