Chapter 9

Find the vertex for each parabola. Then determine a reasonable viewing rectangle on your graphing utility and use it to graph the parabola. $$y=0.01 x^{2}+0.6 x+100$$

The distance, \(d,\) in feet, that an object falls in \(t\) seconds is modeled by the formula \(d=16 t^{2} .\) Use this formula to solve Exercises \(75-76\) If you drop a rock from a cliff 400 feet above the water how long will it take for the rock to hit the water?

For the quadratic equation \(-2 x^{2}+3 x=0,\) we have \(a=-2, b=3,\) and \(c=0\).

Solve each equation or system of equations. $$7(x-2)=10-2(x+3)$$

The distance, \(d,\) in feet, that an object falls in \(t\) seconds is modeled by the formula \(d=16 t^{2} .\) Use this formula to solve Exercises \(75-76\) If you drop a rock from a cliff 576 feet above the water, how long will it take for the rock to hit the water?

Any quadratic equation that can be solved by completing the square can be solved by the quadratic formula.

Solve each equation or system of equations. $$\frac{7}{x+2}+\frac{2}{x+3}=\frac{1}{x^{2}+5 x+6}$$

A square flower bed is to be enlarged by adding 2 meters on each side. If the larger square has an area of 144 square meters, what is the length of the original square? (THE IMAGES CANNOT COPY)

The radicand of the quadratic formula, \(b^{2}-4 a c,\) can be used to determine whether \(a x^{2}+b x+c=0\) has solutions that are rational, irrational, or not real numbers. Explain how this works. Is it possible to determine the kinds of answers that one will obtain to a quadratic equation without actually solving the equation? Explain.

Solve each equation or system of equations. $$\left\\{\begin{array}{l}5 x-3 y=-13 \\ x=2-4 y\end{array}\right.$$

A square flower bed is to be enlarged by adding 3 feet on each side. If the larger square has an area of 169 square feet, what is the length of the original square?

Solve: $$x^{2}+2 \sqrt{3} x-9=0$$

Here are two sets of ordered pairs: $$\begin{array}{l} \text { set } 1:\\{(1,5),(2,5)\\} \\ \text { set } 2:\\{(5,1),(5,2)\\} \end{array}$$ In which set is each \(x\) -coordinate paired with one and only one \(y\) -coordinate?

A machine produces open boxes using square sheets of metal. The figure illustrates that the machine cuts equa sized squares measuring 2 inches on a side from the corners and then shapes the metal into an open box by turning up the sides. If each box must have a volume of 200 cubic inches, find the size of the length and width of the open box. (THE IMAGES CANNOT COPY)

A rectangular vegetable garden is 5 feet wide and 9 feet long. The garden is to be surrounded by a tile border of uniform width. If there are 40 square feet of tile for the border, how wide, to the nearest tenth of a foot, should it be?

A machine produces open boxes using square sheets of metal. The machine cuts equal-sized squares measuring 3 inches on a side from the corners and then shapes the metal into an open box by turning up the sides. If each box must have a volume of 75 cubic inches, find the size of the length and width of the open box.

Graph the formula in Exercise 53 , $$y=-16 x^{2}+60 x+4$$ in a \([0,4,1]\) by \([0,65,5]\) viewing rectangle. Use the graph to verify your solution to the exercise.

Evaluate \(x^{2}+3 x+5\) for \(x=-3\)

What is the square root property?

Explain how to solve \((x-1)^{2}=16\) using the square root property.

Evaluate: \(125^{-4}\). (Section 8.6, Example 3)

In your own words, state the Pythagorean Theorem.

Rationalize the denominator: \(\frac{12}{3+\sqrt{5}}\) (Section 8.4, Example 3)

In the 1939 movie The Wizard of \(O z,\) upon being presented with a Th.D. (Doctor of Thinkology), the Scarecrow proudly exclaims, "The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side." Did the Scarecrow get the Pythagorean Theorem right? In particular, describe four errors in the Scarecrow's statement. (THE IMAGES CANNOT COPY)

Multiply: \(\quad(x-y)\left(x^{2}+x y+y^{2}\right)\) (Section \(5.2,\) Example 7 )

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I have a graphing calculator, so I used the \(x\) -coordinates of the intersection points of the graphs of \(y_{1}=(x-2)^{2}\) and \(y_{2}=25\) to verify the solution set for \((x-2)^{2}=25\)

Can squaring a real number result in a negative number? Based on your answer, are \(\sqrt{-1}\) and \(\sqrt{-4}\) real numbers?

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I've noticed that in mathematics there is a connection between topics, such as using the Pythagorean Theorem to derive the distance formula.

Llist the numbers from each set that are: (A). rational numbers; (B). irrational numbers; (C). real numbers; (D). not real numbers. (Hint: Your answer to each question in Exercise 85 should be "no." $$[-\sqrt{9},-\sqrt{7}, \sqrt{-9}, \sqrt{-7}, \sqrt{0}, \sqrt{7}, \sqrt{9}]$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. When I use the square root property to determine the length of a right triangle's side, I don't even bother to list the negative square root.

Determine whether each statement is true or false If the statement is false, make the necessary change(s) to produce a true statement. The equation \((x+5)^{2}=8\) is equivalent to \(x+5=2 \sqrt{2}\).

Determine whether each statement is true or false If the statement is false, make the necessary change(s) to produce a true statement. The equation \(x^{2}=-1\) has no solutions that are real numbers.

Determine whether each statement is true or false If the statement is false, make the necessary change(s) to produce a true statement. The solutions of \(3 x^{2}-5=0\) are \(\frac{\sqrt{5}}{3}\) and \(-\frac{\sqrt{5}}{3}\)

Find the value(s) of \(x\) if the distance between \((-3,-2)\) and \((x,-5)\) is 5 units.

Use a graphing utility to solve \(4-(x+1)^{2}=0 .\) Graph \(y=4-(x+1)^{2}\) in a \([-5,5,1]\) by \([-5,5,1]\) viewing rectangle. The equation's solutions are the graph's \(x\) -intercepts. Check by substitution in the given equation.

Use a graphing utility to solve \((x-1)^{2}-9=0\) Graph \(y=(x-1)^{2}-9\) in a \([-5,5,1]\) by \([-9,3,1]\) viewing rectangle. The equation's solutions are the graph's \(x\) -intercepts. Check by substitution in the given equation.

Factor completely: \(12 x^{2}+14 x-6\) (Section 6.5, Example 2)

$$\text { Divide: } \frac{x^{2}-x-6}{3 x-3} \div \frac{x^{2}-4}{x-1}$$

Solve: \(4(x-5)=22+2(6 x+3)\) (Section \(2.3,\) Example 3 )

Will help you prepare for the material covered in the next section. $$\text { Factor: } x^{2}+8 x+16$$

Will help you prepare for the material covered in the next section. $$\text { Factor: } x^{2}-14 x+49$$

Will help you prepare for the material covered in the next section. $$\text { Factor: } x^{2}+5 x+\frac{25}{4}$$