Chapter 6

Solve each inequality. $$3 x^{3}+12 x^{2}>0$$

Set up an equation and solve each problem. Find two numbers such that their sum is 10 and their product is 22 .

Use the quadratic formula to solve each of the quadratic equations. Check your solutions by using the sum and product relationships. $$12 x^{2}-73 x+110=0$$

Solve each quadratic equation using the method that seems most appropriate. $$(x+2)(x-7)=10$$

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. For example, $$ \begin{aligned} \sqrt{-3} \sqrt{-8} &=(i \sqrt{3})(i \sqrt{8}) \\ &=i^{2} \sqrt{24} \\ &=(-1) \sqrt{4} \sqrt{6} \\ &=-2 \sqrt{6} \end{aligned} $$ $$\sqrt{-3} \sqrt{-5}$$

Solve each inequality. $$2 x^{3}+4 x^{2} \leq 0$$

Set up an equation and solve each problem. Find two numbers such that their sum is 6 and their product is \(7 .\)

Use the quadratic formula to solve each of the quadratic equations. Check your solutions by using the sum and product relationships. $$6 x^{2}+11 x-255=0$$

Solve each quadratic equation using the method that seems most appropriate. $$(x-3)(x+5)=-7$$

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. For example, $$ \begin{aligned} \sqrt{-3} \sqrt{-8} &=(i \sqrt{3})(i \sqrt{8}) \\ &=i^{2} \sqrt{24} \\ &=(-1) \sqrt{4} \sqrt{6} \\ &=-2 \sqrt{6} \end{aligned} $$ $$\sqrt{-7} \sqrt{-10}$$

Solve each inequality. $$\frac{2 x}{x+3}>4$$

Set up an equation and solve each problem. Suppose that the sum of two whole numbers is 9 , and the sum of their reciprocals is \(\frac{1}{2}\). Find the numbers.

Use the quadratic formula to solve each of the quadratic equations. Check your solutions by using the sum and product relationships. $$-2 x^{2}+4 x-3=0$$

Solve each quadratic equation using the method that seems most appropriate. $$(x-3)^{2}=12$$

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. For example, $$ \begin{aligned} \sqrt{-3} \sqrt{-8} &=(i \sqrt{3})(i \sqrt{8}) \\ &=i^{2} \sqrt{24} \\ &=(-1) \sqrt{4} \sqrt{6} \\ &=-2 \sqrt{6} \end{aligned} $$ $$\sqrt{-9} \sqrt{-6}$$

Solve each inequality. $$\frac{x}{x-1}>2$$

Set up an equation and solve each problem. The difference between two whole numbers is 8 , and the difference between their reciprocals is \(\frac{1}{6}\). Find the two numbers.

Use the quadratic formula to solve each of the quadratic equations. Check your solutions by using the sum and product relationships. $$-2 x^{2}+6 x-5=0$$

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

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. For example, $$ \begin{aligned} \sqrt{-3} \sqrt{-8} &=(i \sqrt{3})(i \sqrt{8}) \\ &=i^{2} \sqrt{24} \\ &=(-1) \sqrt{4} \sqrt{6} \\ &=-2 \sqrt{6} \end{aligned} $$ $$\sqrt{-8} \sqrt{-16}$$

Solve each inequality. $$\frac{x-1}{x-5} \leq 2$$

Set up an equation and solve each problem. The sum of the lengths of the two legs of a right triangle is 21 inches. If the length of the hypotenuse is 15 inches, find the length of each leg.

Use the quadratic formula to solve each of the quadratic equations. Check your solutions by using the sum and product relationships. $$-6 x^{2}+2 x+1=0$$

Solve each quadratic equation using the method that seems most appropriate. $$3 n^{2}-6 n+4=0$$

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. For example, $$ \begin{aligned} \sqrt{-3} \sqrt{-8} &=(i \sqrt{3})(i \sqrt{8}) \\ &=i^{2} \sqrt{24} \\ &=(-1) \sqrt{4} \sqrt{6} \\ &=-2 \sqrt{6} \end{aligned} $$ $$\sqrt{-15} \sqrt{-5}$$

Solve each inequality. $$\frac{x+2}{x+4} \leq 3$$

Set up an equation and solve each problem. The length of a rectangular floor is 1 meter less than twice its width. If a diagonal of the rectangle is 17 meters, find the length and width of the floor.

Use the quadratic formula to solve each of the quadratic equations. Check your solutions by using the sum and product relationships. $$-2 x^{2}+4 x+1=0$$

Solve each quadratic equation using the method that seems most appropriate. $$2 n^{2}-2 n-1=0$$

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. For example, $$ \begin{aligned} \sqrt{-3} \sqrt{-8} &=(i \sqrt{3})(i \sqrt{8}) \\ &=i^{2} \sqrt{24} \\ &=(-1) \sqrt{4} \sqrt{6} \\ &=-2 \sqrt{6} \end{aligned} $$ $$\sqrt{-2} \sqrt{-20}$$

Solve each inequality. $$\frac{x+2}{x-3}>-2$$

Set up an equation and solve each problem. A rectangular plot of ground measuring 12 meters by 20 meters is surrounded by a sidewalk of a uniform width (see Figure 6.9). The area of the sidewalk is 68 square meters. Find the width of the walk.

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?

Solve each quadratic equation using the method that seems most appropriate. $$n(n+8)=240$$

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. For example, $$ \begin{aligned} \sqrt{-3} \sqrt{-8} &=(i \sqrt{3})(i \sqrt{8}) \\ &=i^{2} \sqrt{24} \\ &=(-1) \sqrt{4} \sqrt{6} \\ &=-2 \sqrt{6} \end{aligned} $$ $$\sqrt{-2} \sqrt{-27}$$

Solve each inequality. $$\frac{x-1}{x-2}<-1$$

Set up an equation and solve each problem. A 5-inch by 7-inch picture is surrounded by a frame of uniform width. The area of the picture and frame together is 80 square inches. Find the width of the frame.

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?

Solve each quadratic equation using the method that seems most appropriate. $$t(t-26)=-160$$

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. For example, $$ \begin{aligned} \sqrt{-3} \sqrt{-8} &=(i \sqrt{3})(i \sqrt{8}) \\ &=i^{2} \sqrt{24} \\ &=(-1) \sqrt{4} \sqrt{6} \\ &=-2 \sqrt{6} \end{aligned} $$ $$\sqrt{-3} \sqrt{-15}$$

Solve each inequality. $$\frac{3 x+2}{x+4} \leq 2$$

Set up an equation and solve each problem. The perimeter of a rectangle is 44 inches, and its area is 112 square inches. Find the length and width of the rectangle.

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? 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 quadratic equation using the method that seems most appropriate. $$3 x^{2}+5 x=-2$$

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. For example, $$ \begin{aligned} \sqrt{-3} \sqrt{-8} &=(i \sqrt{3})(i \sqrt{8}) \\ &=i^{2} \sqrt{24} \\ &=(-1) \sqrt{4} \sqrt{6} \\ &=-2 \sqrt{6} \end{aligned} $$ $$\sqrt{6} \sqrt{-8}$$

Solve each inequality. $$\frac{2 x-1}{x+2} \geq-1$$

Set up an equation and solve each problem. A rectangular piece of cardboard is 2 units longer than it is wide. From each of its corners a square piece 2 units on a side is cut out. The flaps are then turned up to form an open box that has a volume of 70 cubic units. Find the length and width of the original piece of cardboard.

Expressing solutions to the nearest one-thousandth. $$x^{2}-6 x-10=0$$

Solve each quadratic equation using the method that seems most appropriate. $$2 x^{2}-7 x=-5$$

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. For example, $$ \begin{aligned} \sqrt{-3} \sqrt{-8} &=(i \sqrt{3})(i \sqrt{8}) \\ &=i^{2} \sqrt{24} \\ &=(-1) \sqrt{4} \sqrt{6} \\ &=-2 \sqrt{6} \end{aligned} $$ $$\sqrt{-75} \sqrt{3}$$