Chapter 12

Find the distance between the two points. Round the result to the nearest hundredth if necessary. $$(3,-1),(0,3)$$

Find the missing length of the right triangle if a and b are the lengths of the legs and c is the length of the hypotenuse. $$a=10, b=24$$

Sketch the graph of the function. $$y=\sqrt{x}+5$$

Choose a method to solve the quadratic equation. What method did you use? Explain your choice. $$x^{2}-9=0$$

Show whether the expression is a solution of the equation. $$x^{2}-8 x+8=0 ; 4+2 \sqrt{2}$$

Solve the equation. Check for extraneous solutions. $$\sqrt{x}+5=0$$

Prove the theorem. (Use the basic axioms of algebra and the definition of subtraction given in Example \(1 .\) ) If \(a\) is any real number, then \(-1(a)=-a\)

Find the distance between the two points. Round the result to the nearest hundredth if necessary. $$(5,8),(-2,3)$$

Sketch the graph of the function. $$y=3 \sqrt{x+1}$$

Find the missing length of the right triangle if a and b are the lengths of the legs and c is the length of the hypotenuse. $$b=9, c=16$$

Choose a method to solve the quadratic equation. What method did you use? Explain your choice. $$-3 x^{2}+5 x+5=0$$

Solve the equation. Check for extraneous solutions. $$\sqrt{x}-10=0$$

A student proposes the following conjecture. The sum of the first n odd integers is \(n^{2} .\) She gives four examples: \(1=1^{2}\) \(1+3=4=2^{2}, 1+3+5=9=3^{2},\) and \(1+3+5+7=16=4^{2} .\) Do the examples prove her conjecture? Explain. Do you think the conjecture is true?

Find the distance between the two points. Round the result to the nearest hundredth if necessary. 18\. $$(-3,1),(2,6)$$

For a fire hose with a nozzle that has a diameter of 2 inches, the flow rate \(f\) (in gallons per minute) can be modeled by \(f=120 \sqrt{p}\) where \(p\) is the nozzle pressure in pounds per square inch. Sketch a graph of the model.

Find the missing length of the right triangle if a and b are the lengths of the legs and c is the length of the hypotenuse. $$a=14, c=21$$

Choose a method to solve the quadratic equation. What method did you use? Explain your choice. $$x^{2}+2 x-14=0$$

Simplify the expression. $$5 \sqrt{7}+2 \sqrt{7}$$

Solve the equation. Check for extraneous solutions. $$\sqrt{x}-15=0$$

Find a counterexample to show that the statement is not true. If \(a\) and \(b\) are real numbers, then \((a+b)^{2}=a^{2}+b^{2}\)

Find the distance between the two points. Round the result to the nearest hundredth if necessary. $$(-6,-2),(-3,-5)$$

For a fire hose with a nozzle that has a diameter of 2 inches, the flow rate \(f\) (in gallons per minute) can be modeled by \(f=120 \sqrt{p}\) where \(p\) is the nozzle pressure in pounds per square inch. If the flow rate is 1200 gallons per minute, what is the nozzle pressure?

Choose a method to solve the quadratic equation. What method did you use? Explain your choice. $$3 x^{2}-2=0$$

Simplify the expression. $$\sqrt{3}+5 \sqrt{3}$$

Solve the equation. Check for extraneous solutions. $$\sqrt{x}-0=0$$

SLOUD HEIGHT You are doing an experiment for your science class. You shine a spotlight from the ground straight up onto a cloud to measure the height of the cloud. Your friend stands 500 feet from the spotlight. She estimates that the angle formed between the ground and the line from her feet to the cloud is \(25^{\circ} .\) Draw a sketch to model the situation. Then find the height of the cloud.

Find a counterexample to show that the statement is not true. If \(a, b,\) and \(c\) are nonzero real numbers, then \((a \div b) \div c=a \div(b \div c)\)

Find the distance between the two points. Round the result to the nearest hundredth if necessary. $$(4,5),(-1,3)$$

Evaluate the function for the given value of \(x .\) Round your answer to the nearest tenth. $$y=3 \sqrt{x} ; 9$$

Find the term that should be added to the expression to create a perfect square trinomial. $$x^{2}-12 x$$

Simplify the expression. $$11 \sqrt{3}-12 \sqrt{3}$$

Solve the equation. Check for extraneous solutions. $$\sqrt{6 x}-13=23$$

SoLLINSVILLE CATSUP BOTTLE You are standing 197 feet from the base of the world's largest catsup bottle located in Collinsville, Illinois. You estimate that the angle between your eye level and the line from your eyes to the top of the bottle is \(40^{\circ} .\) If you are 5 feet tall, about how high is the top of the bottle? (GRAPH CANT COPY)

Find a counterexample to show that the statement is not true. If \(a\) and \(b\) are integers, then \(a \div b\) is an integer.

Find the distance between the two points. Round the result to the nearest hundredth if necessary. $$(-6,1),(3,1)$$

Evaluate the function for the given value of \(x .\) Round your answer to the nearest tenth. $$y=\frac{1}{2} \sqrt{x}-1 ; 16$$

Simplify the expression. $$2 \sqrt{6}-\sqrt{6}$$

Solve the equation. Check for extraneous solutions. $$\sqrt{4 x+1}+5=10$$

Find the distance between the two points. Round the result to the nearest hundredth if necessary. $$(-2,-1),(3,-3)$$

Evaluate the function for the given value of \(x .\) Round your answer to the nearest tenth. $$y=\sqrt{x-7} ; 15$$

Find the term that should be added to the expression to create a perfect square trinomial. $$x^{2}+21 x$$

Simplify the expression. $$\sqrt{32}+\sqrt{2}$$

Solve the equation. Check for extraneous solutions. $$\sqrt{9-x}-10=14$$

Find the distance between the two points. Round the result to the nearest hundredth if necessary. $$(3.5,6),(-3.5,-2)$$

Evaluate the function for the given value of \(x .\) Round your answer to the nearest tenth. $$y=\sqrt{3 x-5} ; 7$$

Find the term that should be added to the expression to create a perfect square trinomial. $$x^{2}-22 x$$

Simplify the expression. $$\sqrt{75}+\sqrt{3}$$

Solve the equation. Check for extraneous solutions. $$\sqrt{5 x+1}+2=6$$

Use an indirect proof to prove that the conclusion is true. Your bus leaves a track meet at 4: 30 P.M. and does not travel faster than 60 miles per hour. The meet is 45 miles from home. Your bus will not get you home in time for dinner at \(5 \mathrm{P.M}\).

Find the distance between the two points. Round the result to the nearest hundredth if necessary. $$\left(\frac{1}{2}, \frac{1}{4}\right),(2,1)$$