Chapter 1

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement . $$(-\infty, 3) \cup(-\infty,-2)=(-\infty,-2)$$

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 length and width of the open box.

A piece of wire is 8 inches long. The wire is cut into two pieces and then each piece is bent into a square. Find the length of each piece if the sum of the areas of these squares is to be 2 square inches.

What's wrong with this argument? Suppose \(x\) and \(y\) represent two real numbers, where \(x>y\). \(2>1 \quad\) This is a true statement \(2(y-x)>1(y-x) \quad\) Multiply both sides by \(y-x\) \(2 y-2 x>y-x \quad\) Use the distributive property. \(y-2 x>-x \quad\) Subtract \(y\) from both sides. \(y>x \quad\) Add \(2 x\) to both sides. The final inequality, \(y>x,\) is impossible because we were initially given \(x>y\).

What is a quadratic equation?

Explain how to solve \(x^{2}+6 x+8=0\) using factoring and the zero-product principle.

Each group member should research one situation that provides two different pricing options. These can involve areas such as public transportation options (with or without discount passes), cellphone plans, long-distance telephone plans, or anything of interest. Be sure to bring in all the details for each option. At a second group meeting, select the two pricing situations that are most interesting and relevant. Using each situation, write a word problem about selecting the better of the two options. The word problem should be one that can be solved using a linear inequality. The group should turn in the two problems and their solutions.

Explain how to solve \(x^{2}+6 x+8=0\) by completing the square.

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

Explain how to solve \(x^{2}+6 x+8=0\) using the quadratic formula.

Graph \(y=2 x\) and \(y-2 x+4\) in the same rectangular coordinate system. Select integers for \(x,\) starting with \(-2\) and ending with 2.

How is the quadratic formula derived?

What is the discriminant and what information does it provide about a quadratic equation?

If you are given a quadratic equation, how do you determine which method to use to solve it?

Describe the relationship between the real solutions of \(a x^{2}+b x+c=0\) and the graph of \(y=a x^{2}+b x+c\).

If a quadratic equation has imaginary solutions, how is this shown on the graph of \(y=a x^{2}+b x+c ?\)

In Exercises \(166-169\), determine whether each statement makes sense or does not make sense, and explain your reasoning. Because I want to solve \(25 x^{2}-169=0\) fairly quickly, I'll use the quadratic formula.

In Exercises \(166-169\), determine whether each statement makes sense or does not make sense, and explain your reasoning. I obtained \(-17\) for the discriminant, so there are two imaginary irrational solutions.

In Exercises \(170-173\), determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Any quadratic equation that can be solved by completing the square can be solved by the quadratic formula.

Write a quadratic equation in general form whose solution set is \(\\{-3,5\\}\).

Solve for \(t: s=-16 t^{2}+v_{0} t\)

A rectangular swimming pool is 12 meters long and 8 meters wide. A tile border of uniform width is to be built around the pool using 120 square meters of tile. The tile is from a discontinued stock (so no additional materials are available) and all 120 square meters are to be used. How wide should the border be? Round to the nearest tenth of a meter. If zoning laws require at least a 2 -meter-wide border around the pool, can this be done with the available tile?

Exercises \(177-179\) will help you prepare for the material covered in the next section. Factor completely: \(x^{3}+x^{2}-4 x-4\)

Exercises \(177-179\) will help you prepare for the material covered in the next section. Use the special product \((A+B)^{2}=A^{2}+2 A B+B^{2}\) to multiply: \((\sqrt{x+4}+1)^{2}\)

Exercises \(177-179\) will help you prepare for the material covered in the next section. If \(-8\) is substituted for \(x\) in the equation \(5 x^{\frac{2}{3}}+11 x^{\frac{1}{3}}+2=0\) is the resulting statement true or false?