Chapter 6

Why is the solution set for \((x-2)^{2} \geq 0\) the set of all real numbers?

Set up an equation and solve each problem. Suppose that Arlene can mow the entire lawn in 40 minutes less time with the power mower than she can with the push mower. One day the power mower broke down after she had been mowing for 30 minutes. She finished the lawn with the push mower in 20 minutes. How long does it take Arlene to mow the entire lawn with the power mower?

For each quadratic equation, first use the discriminant to determine whether the equation has two nonreal complex solutions, one real solution with a multiplicity of two, or two real solutions. Then solve the equation. $$ 2 x^{2}-6 x=-1 $$

Solve each quadratic equation using the method that seems most appropriate. $$ 5(x+2)^{2}+1=16 $$

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. $$ \frac{\sqrt{-96}}{\sqrt{2}} $$

Set up an equation and solve each problem. A student did a word processing job for \(\$ 24\). It took him 1 hour longer than he expected, and therefore he earned \(\$ 4\) per hour less than he anticipated. How long did he expect that it would take to do the job?

Your friend states that the equation \(-2 x^{2}+4 x-1=0\) must be changed to \(2 x^{2}-4 x+1=0\) (by multiplying both sides by \(-1\) ) before the quadratic formula can be applied. Is she right about this? If not, how would you convince her she is wrong?

Use the method of completing the square to solve \(a x^{2}+\) \(b x+c=0\) for \(x\), where \(a, b\), and \(c\) are real numbers and \(a \neq 0\).

Find each of the products and express the answers in the standard form of a complex number. $$ (5 i)(4 i) $$

The product \((x-2)(x+3)\) is positive if both factors are negative or if both factors are positive. Therefore, we can solve \((x-2)(x+3)>0\) as follows: \((x-2<0\) and \(x+3<0)\) or \((x-2>0\) and \(x+3>0)\) \((x<2\) and \(x<-3)\) or \((x>2\) and \(x>-3)\) $$ x<-3 \text { or } x>2 $$ The solution set is \((-\infty,-3) \cup(2, \infty)\). Use this type of analysis to solve each of the following. (a) \((x-2)(x+7)>0\) (b) \((x-3)(x+9) \geq 0\) (c) \((x+1)(x-6) \leq 0\) (d) \((x+4)(x-8)<0\) (e) \(\frac{x+4}{x-7}>0\) (f) \(\frac{x-5}{x+8} \leq 0\)

Set up an equation and solve each problem. A group of students agreed that each would chip in the same amount to pay for a party that would cost \(\$ 100\). Then they found 5 more students interested in the party and in sharing the expenses. This decreased the amount each had to pay by \(\$ 1\). How many students were involved in the party and how much did each student have to pay?

Another of your friends claims that the quadratic formula can be used to solve the equation \(x^{2}-9=0\). How would you react to this claim?

Explain the process of completing the square to solve a quadratic equation.

Find each of the products and express the answers in the standard form of a complex number. $$ (-6 i)(9 i) $$

Set up an equation and solve each problem. A group of students agreed that each would contribute the same amount to buy their favorite teacher an \(\$ 80\) birthday gift. At the last minute, 2 of the students decided not to chip in. This increased the amount that the remaining students had to pay by \(\$ 2\) per student. How many students actually contributed to the gift?

Why must we change the equation \(3 x^{2}-2 x=4\) to \(3 x^{2}-2 x-4=0\) before applying the quadratic formula?

Give a step-by-step description of how to solve \(3 x^{2}+9 x-4=0\) by completing the square.

For Problems \(63-68, a\) and \(b\) represent the lengths of the legs of a right triangle, and \(c\) represents the length of the hypotenuse. Express answers in simplest radical form. Find \(c\) if \(a=4\) centimeters and \(b=6\) centimeters.

Find each of the products and express the answers in the standard form of a complex number. $$ (7 i)(-6 i) $$

Set up an equation and solve each problem. The formula \(D=\frac{n(n-3)}{2}\) yields the number of diagonals, \(D\), in a polygon of \(n\) sides. Find the number of sides of a polygon that has 54 diagonals.

The solution set for \(x^{2}-4 x-37=0\) is \(\\{2 \pm \sqrt{41}\\}\). With a calculator, we found a rational approximation, to the nearest one-thousandth, for each of these solutions. $$ 2-\sqrt{41}=-4.403 \quad \text { and } \quad 2+\sqrt{41}=8.403 $$ Thus the solution set is \(\\{-4.403,8.403\\}\), with the answers rounded to the nearest one-thousandth. Solve each of the equations in Problems 64-73, expressing solutions to the nearest one-thousandth. $$ x^{2}-6 x-10=0 $$

Solve Problems 64-67 for the indicated variable. Assume that all letters represent positive numbers. $$ \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \quad \text { for } y $$

For Problems \(63-68, a\) and \(b\) represent the lengths of the legs of a right triangle, and \(c\) represents the length of the hypotenuse. Express answers in simplest radical form. Find \(c\) if \(a=3\) meters and \(b=7\) meters.

Find each of the products and express the answers in the standard form of a complex number. $$ (-5 i)(-12 i) $$

Set up an equation and solve each problem. The formula \(S=\frac{n(n+1)}{2}\) yields the sum, \(S\), of the first \(n\) natural numbers \(1,2,3,4, \ldots\) How many consecutive natural numbers starting with 1 will give a sum of 1275 ?

The solution set for \(x^{2}-4 x-37=0\) is \(\\{2 \pm \sqrt{41}\\}\). With a calculator, we found a rational approximation, to the nearest one-thousandth, for each of these solutions. $$ 2-\sqrt{41}=-4.403 \quad \text { and } \quad 2+\sqrt{41}=8.403 $$ Thus the solution set is \(\\{-4.403,8.403\\}\), with the answers rounded to the nearest one-thousandth. Solve each of the equations in Problems 64-73, expressing solutions to the nearest one-thousandth. $$ x^{2}-16 x-24=0 $$

Solve Problems 64-67 for the indicated variable. Assume that all letters represent positive numbers. $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \quad \text { for } x $$

For Problems \(63-68, a\) and \(b\) represent the lengths of the legs of a right triangle, and \(c\) represents the length of the hypotenuse. Express answers in simplest radical form. Find \(a\) if \(c=12\) inches and \(b=8\) inches.

Find each of the products and express the answers in the standard form of a complex number. $$ (3 i)(2-5 i) $$

The solution set for \(x^{2}-4 x-37=0\) is \(\\{2 \pm \sqrt{41}\\}\). With a calculator, we found a rational approximation, to the nearest one-thousandth, for each of these solutions. $$ 2-\sqrt{41}=-4.403 \quad \text { and } \quad 2+\sqrt{41}=8.403 $$ Thus the solution set is \(\\{-4.403,8.403\\}\), with the answers rounded to the nearest one-thousandth. Solve each of the equations in Problems 64-73, expressing solutions to the nearest one-thousandth. $$ x^{2}+6 x-44=0 $$

Solve Problems 64-67 for the indicated variable. Assume that all letters represent positive numbers. $$ s=\frac{1}{2} g t^{2} \quad \text { for } t $$

For Problems \(63-68, a\) and \(b\) represent the lengths of the legs of a right triangle, and \(c\) represents the length of the hypotenuse. Express answers in simplest radical form. Find \(a\) if \(c=8\) feet and \(b=6\) feet.

Find each of the products and express the answers in the standard form of a complex number. $$ (7 i)(-9+3 i) $$

Set up an equation and solve each problem. Suppose that \(\$ 500\) is invested at a certain rate of interest compounded annually for 2 years. If the accumulated value at the end of 2 years is \(\$ 594.05\), find the rate of interest.

The solution set for \(x^{2}-4 x-37=0\) is \(\\{2 \pm \sqrt{41}\\}\). With a calculator, we found a rational approximation, to the nearest one-thousandth, for each of these solutions. $$ 2-\sqrt{41}=-4.403 \quad \text { and } \quad 2+\sqrt{41}=8.403 $$ Thus the solution set is \(\\{-4.403,8.403\\}\), with the answers rounded to the nearest one-thousandth. Solve each of the equations in Problems 64-73, expressing solutions to the nearest one-thousandth. $$ x^{2}+10 x-46=0 $$

Solve Problems 64-67 for the indicated variable. Assume that all letters represent positive numbers. $$ A=\pi r^{2} \quad \text { for } r $$

For Problems \(63-68, a\) and \(b\) represent the lengths of the legs of a right triangle, and \(c\) represents the length of the hypotenuse. Express answers in simplest radical form. Find \(b\) if \(c=17\) yards and \(a=15\) yards.

Find each of the products and express the answers in the standard form of a complex number. $$ (-6 i)(-2-7 i) $$

Set up an equation and solve each problem. Suppose that \(\$ 10,000\) is invested at a certain rate of interest compounded annually for 2 years. If the accumulated value at the end of 2 years is \(\$ 12,544\), find the rate of interest.

The solution set for \(x^{2}-4 x-37=0\) is \(\\{2 \pm \sqrt{41}\\}\). With a calculator, we found a rational approximation, to the nearest one-thousandth, for each of these solutions. $$ 2-\sqrt{41}=-4.403 \quad \text { and } \quad 2+\sqrt{41}=8.403 $$ Thus the solution set is \(\\{-4.403,8.403\\}\), with the answers rounded to the nearest one-thousandth. Solve each of the equations in Problems 64-73, expressing solutions to the nearest one-thousandth. $$ x^{2}+8 x+2=0 $$

Solve each of the following equations for \(x\). $$ x^{2}+8 a x+15 a^{2}=0 $$

For Problems \(63-68, a\) and \(b\) represent the lengths of the legs of a right triangle, and \(c\) represents the length of the hypotenuse. Express answers in simplest radical form. Find \(b\) if \(c=14\) meters and \(a=12\) meters.

Find each of the products and express the answers in the standard form of a complex number. $$ (-9 i)(-4-5 i) $$

How would you solve the equation \(x^{2}-4 x=252\) ? Explain your choice of the method that you would use.

The solution set for \(x^{2}-4 x-37=0\) is \(\\{2 \pm \sqrt{41}\\}\). With a calculator, we found a rational approximation, to the nearest one-thousandth, for each of these solutions. $$ 2-\sqrt{41}=-4.403 \quad \text { and } \quad 2+\sqrt{41}=8.403 $$ Thus the solution set is \(\\{-4.403,8.403\\}\), with the answers rounded to the nearest one-thousandth. Solve each of the equations in Problems 64-73, expressing solutions to the nearest one-thousandth. $$ x^{2}+9 x+3=0 $$

Solve each of the following equations for \(x\). $$ x^{2}-5 a x+6 a^{2}=0 $$

Find each of the products and express the answers in the standard form of a complex number. $$ (3+2 i)(5+4 i) $$

Explain how you would solve \((x-2)(x-7)=0\) and also how you would solve \((x-2)(x-7)=4\).

The solution set for \(x^{2}-4 x-37=0\) is \(\\{2 \pm \sqrt{41}\\}\). With a calculator, we found a rational approximation, to the nearest one-thousandth, for each of these solutions. $$ 2-\sqrt{41}=-4.403 \quad \text { and } \quad 2+\sqrt{41}=8.403 $$ Thus the solution set is \(\\{-4.403,8.403\\}\), with the answers rounded to the nearest one-thousandth. Solve each of the equations in Problems 64-73, expressing solutions to the nearest one-thousandth. $$ 4 x^{2}-6 x+1=0 $$

Find each of the products and express the answers in the standard form of a complex number. $$ (4+3 i)(6+i) $$