Chapter 6

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.

Use the discriminant to help solve each problem. Determine \(k\) so that \(3 x^{2}-k x-2=0\) has real solutions.

For the indicated variable. Assume that all letters represent positive numbers. \(s=\frac{1}{2} g t^{2}\) for \(t\)

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. 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 ?

For the indicated variable. Assume that all letters represent positive numbers. \(A=\pi r^{2}\) for \(r\)

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. At a point 16 yards from the base of a tower, the distance to the top of the tower is 4 yards more than the height of the tower (see Figure 6.10). Find the height of the tower.

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

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

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.

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

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.

Solve each of the following equations for \(x\). $$10 x^{2}-31 a x-14 a^{2}=0$$

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

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

\(a\) 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. $$(6-2 i)(7-i)$$

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

Solve each of the following equations for \(x\). $$4 x^{2}+4 b x+b^{2}=0$$

\(a\) 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. $$(8-4 i)(7-2 i)$$

One of our problem-solving suggestions is to look for a guideline that can be used to help determine an equation. What does this suggestion mean to you?

\(a\) 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. $$(-3-2 i)(5+6 i)$$

Can a quadratic equation with integral coefficients have exactly one nonreal complex solution? Explain your answer.

\(a\) 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. $$(-5-3 i)(2-4 i)$$

Solve each equation. $$x-9 \sqrt{x}+18=0$$

\(a\) 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. $$(9+6 i)(-1-i)$$

Solve each equation. $$x-4 \sqrt{x}+3=0$$

\(a\) 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. $$(10+2 i)(-2-i)$$

Solve each equation. $$x+\sqrt{x}-2=0$$

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

Solve each equation. $$x^{\frac{2}{3}}+x^{\frac{1}{3}}-6=0$$

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

Solve each equation. $$6 x^{\frac{2}{3}}-5 x^{\frac{1}{3}}-6=0$$

Find each of the products and express the answers in the standard form of a complex number. $$(-2-4 i)^{2}$$

Solve each equation. $$x^{-2}+4 x^{-1}-12=0$$

Find each of the products and express the answers in the standard form of a complex number. $$(-3-6 i)^{2}$$

Solve each equation. $$12 x^{-2}-17 x^{-1}-5=0$$

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

Solve each equation. $$(5 x+6)^{\frac{1}{2}}=x$$

Find each of the products and express the answers in the standard form of a complex number. $$(5-7 i)(5+7 i)$$

Solve each equation. $$(3 x+4)^{\frac{1}{2}}=x$$

Find each of the products and express the answers in the standard form of a complex number. $$(-1+2 i)(-1-2 i)$$

Solve each equation. $$x^{\frac{2}{3}}=2$$

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