Chapter 5

In \(26-37,\) find each product. $$ (3+2 i)(3+3 i) $$

In \(19-34,\) write each sum or difference in terms of \(i\) $$ \sqrt{4}+\sqrt{-32}+\sqrt{-8} $$

Based on data from a local college, a statistician determined that the average student's grade point average \((\mathrm{GPA})\) at this college is a function of the number of hours, \(h,\) he or she studies each weck. The grade point average can be estimated by the function \(\mathrm{G}(h)=0.006 h^{2}+0.02 h+1.2\) for \(0 \leq h \leq 20\) . a. To the nearest tenth, what is the average GPA of a student who does no studying? b. To the nearest tenth, what is the average GPA of a student who studies 12 hours per week? c. To the nearest tenth, how many hours of studying are required for the average student to achieve a 3.2 GPA?

a. Complete the square to find the roots of the equation \(x^{2}-5 x+1=0\) . b. Write, to the nearest tenth, a rational approximation for the roots.

In \(18-35,\) find each common solution algebraically. Express irrational roots in simplest radical form. $$ \begin{array}{l}{y=x^{2}+4 x+4} \\ {y=4 x+6}\end{array} $$

In \(19-28 :\) a. Find \(\mathrm{f}(a)\) for each given function. b. Is \(a\) a root of the function? $$ \mathrm{f}(x)=-5 x^{3}+5 x^{2}+2 x+3 \text { and } a=\frac{3}{2} i $$

One root of the equation \(-3 x^{2}+9 x+c=0\) is \(\sqrt{2}\) a. Find the other root. b. Find the value of c. c. Explain why the roots of this equation are not conjugates.

In \(26-37,\) find each product. $$ (2-3 i)^{2} $$

In \(19-34,\) write each sum or difference in terms of \(i\) $$ 3+\sqrt{-28}-7-\sqrt{-7} $$

In \(18-35,\) find each common solution algebraically. Express irrational roots in simplest radical form. $$ \begin{array}{l}{2 x^{2}+x-y+1=0} \\ {x-y+7=0}\end{array} $$

a. Verify by multiplication that \((x-1)\left(x^{2}+x+1\right)=x^{3}-1\) b. Use the factors of \(x^{3}-1\) to find the three roots of \(f(x)=x^{3}-1\) c. If \(x^{3}-1=0,\) then \(x^{3}=1\) and \(x=\sqrt[3]{1}\) . Use the answer to part b to write the three cube roots of \(1 .\) Explain your reasoning. d. Verify that each of the two imaginary roots of \(f(x)=x^{3}-1\) is a cube root of \(1 .\)

One root of the equation \(-x^{2}-11 x+c=0\) is \(\sqrt{3}\) a. Find the other root. b. Find the value of c. c. Explain why the roots of this equation are not conjugates.

The height, in feet, to which of a golf ball rises when shot upward from ground level is described by the function h(t) \(=-16 t^{2}+48 t\) where \(t\) is the time elapsed in seconds. Use the discriminant to determine if the golf ball will reach a height of 32 feet.

In \(26-37,\) find each product. $$ (4-5 i)(3-2 i) $$

In \(19-34,\) write each sum or difference in terms of \(i\) $$ -3-\sqrt{-10}+2-\sqrt{-90} $$

In \(28-33,\) without graphing the parabola, describe the translation, reflection, and \(/\) or scaling that must be applied to \(y=x^{2}\) to obtain the graph of each given function. $$ \mathrm{f}(x)=x^{2}+2 x-2 $$

Write a quadratic equation with integer coefficients for each pair of roots. \(2,5\)

In \(18-35,\) find each common solution algebraically. Express irrational roots in simplest radical form. $$ \begin{array}{l}{y=x^{2}+2} \\ {2 x-y=-3}\end{array} $$

a. Verify by multiplication that \((x+1)\left(x^{2}-x+1\right)=x^{3}+1\) b. Use the factors of \(x^{3}+1\) to find the three roots of \(\mathrm{f}(x)=x^{3}+1\) c. If \(x^{3}+1=0,\) then \(x^{3}=-1\) and \(x=\sqrt[3]{-1}\) Use the answer to part \(\mathbf{b}\) to write the three cube roots of \(-1 .\) Explain your reasoning. d. Verify that each of the two imaginary roots of \(f(x)=x^{3}+1\) is a cube root of \(-1\)

The profit function for a company that manufactures cameras is \(\mathrm{P}(x)=-x^{2}+350 x-15,000 .\) Under present conditions, can the company achieve a profit of \(\$ 20,000 ?\) Use the discriminant to explain your answer.

In \(26-37,\) find each product. $$ (7+i)(2+3 i) $$

In \(19-34,\) write each sum or difference in terms of \(i\) $$ -2+\sqrt{-16}+7-\sqrt{-49} $$

In \(28-33,\) without graphing the parabola, describe the translation, reflection, and \(/\) or scaling that must be applied to \(y=x^{2}\) to obtain the graph of each given function. $$ f(x)=x^{2}-6 x-7 $$

Write a quadratic equation with integer coefficients for each pair of roots. \(4,7\)

In \(18-35,\) find each common solution algebraically. Express irrational roots in simplest radical form. $$ \begin{array}{l}{y=2 x^{2}+3 x+4} \\ {y=11 x+6}\end{array} $$

Let \(\mathrm{f}(x)=x^{3}+3 x^{2}-2 x-6\) and \(\mathrm{g}(x)=2 \mathrm{f}(x)=2 x^{3}+6 x-4 x-12\) be two cubic polynomial functions. a. How does the graph of \(\mathrm{f}(x)\) compare with the graph of \(\mathrm{g}(x) ?\) b. How do the roots of \(\mathrm{f}(x)\) compare with the roots of \(\mathrm{g}(x) ?\) c. In general, if \(\mathrm{p}(x)=a \mathrm{q}(x)\) and \(a>0,\) how does the graph of \(\mathrm{p}(x)\) compare with the graph of \(\mathrm{q}(x) ?\) d. How do the roots of \(\mathrm{p}(x)\) compare with the roots of \(\mathrm{q}(x) ?\)

In \(26-37,\) find each product. $$ (-2-i)(1+2 i) $$

In \(19-34,\) write each sum or difference in terms of \(i\) $$ 3+\sqrt{-36}-6+\sqrt{-1} $$

Write a quadratic equation with integer coefficients for each pair of roots. \(-3,4\)

In \(26-37,\) find each product. $$ (-4-i)(-4+i) $$

In \(19-34,\) write each sum or difference in terms of \(i\) $$ \frac{1}{7}+\sqrt{-\frac{1}{8}}+\frac{2}{7}-\sqrt{-\frac{1}{2}} $$

In \(28-33,\) without graphing the parabola, describe the translation, reflection, and \(/\) or scaling that must be applied to \(y=x^{2}\) to obtain the graph of each given function. $$ \mathrm{f}(x)=-x^{2}+x+2 $$

Write a quadratic equation with integer coefficients for each pair of roots. \(-2,-1\)

In \(18-35,\) find each common solution algebraically. Express irrational roots in simplest radical form. $$ \begin{array}{l}{x^{2}+y^{2}=24} \\ {x+y=8}\end{array} $$

In \(26-37,\) find each product. $$ (-12-2 i)(12-2 i) $$

In \(19-34,\) write each sum or difference in terms of \(i\) $$ -\frac{1}{2}+\sqrt{-\frac{2}{3}}-\frac{1}{2}+\sqrt{-\frac{24}{9}} $$

In \(28-33,\) without graphing the parabola, describe the translation, reflection, and \(/\) or scaling that must be applied to \(y=x^{2}\) to obtain the graph of each given function. $$ f(x)=3 x^{2}+6 x+3 $$

Write a quadratic equation with integer coefficients for each pair of roots. \(-3,3\)

In \(18-35,\) find each common solution algebraically. Express irrational roots in simplest radical form. $$ \begin{array}{l}{x^{2}+y^{2}=20} \\ {3 x-y=10}\end{array} $$

In \(26-37,\) find each product. $$ (5-3 i)(5+3 i) $$

In \(19-34,\) write each sum or difference in terms of \(i\) $$ \sqrt{0.2025}+\sqrt{-0.09} $$

Determine the coordinates of the vertex and the equation of the axis of symmetry of \(\mathrm{f}(x)=x^{2}+8 x+5\) by writing the equation in the form \(\mathrm{f}(x)=(x-h)^{2}+k .\) Justify your answer.

Write a quadratic equation with integer coefficients for each pair of roots. \(\frac{1}{2}, \frac{7}{2}\)

In \(18-35,\) find each common solution algebraically. Express irrational roots in simplest radical form. $$ \begin{array}{l}{x^{2}+y^{2}=16} \\ {y=2 x}\end{array} $$

In \(26-37,\) find each product. $$ (3-i)\left(\frac{3}{10}+\frac{1}{10} i\right) $$

In \(35-43,\) write each number in simplest form. $$ 3 i+2 i^{3} $$

The length of a rectangle is 4 feet more than twice the width. The area of the rectangle is 38 square feet. a. Find the dimensions of the rectangle in simplest radical form. b. Show that the product of the length and width is equal to the area. c. Write, to the nearest tenth, rational approximations for the length and width.

Write a quadratic equation with integer coefficients for each pair of roots. \(\frac{3}{4},-\frac{3}{8}\)

In \(26-37,\) find each product. $$ \left(\frac{1}{5}-\frac{1}{10} i\right)(4+2 i) $$

In \(35-43,\) write each number in simplest form. $$ 5 i^{2}+2 i^{4} $$