Chapter 1

The given equation involves a power of the variable. Find all real solutions of the equation. \(3(x-5)^{2}=15\)

Find all solutions of the equation, and express them in the form \(a+b i\) $$ x^{2}+2 x+5=0 $$

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ \frac{2 x+1}{x-5} \leq 3 $$

Solve the equation for the indicated variable. $$ A=2 x^{2}+4 x h ; \quad \text { for } x $$

\(61-70\) . Find all solutions, real and complex, of the equation. $$ x^{3}=1 $$

The given equation involves a power of the variable. Find all real solutions of the equation. \(x^{3}=27\)

Find all solutions of the equation, and express them in the form \(a+b i\) $$ x^{2}-6 x+10=0 $$

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ \frac{3+x}{3-x} \geq 1 $$

Solve the equation for the indicated variable. $$ A=2 \pi r^{2}+2 \pi r h ; \quad \text { for } r $$

\(61-70\) . Find all solutions, real and complex, of the equation. $$ x^{4}-16=0 $$

The given equation involves a power of the variable. Find all real solutions of the equation. \(x^{5}+32=0\)

Find all solutions of the equation, and express them in the form \(a+b i\) $$ x^{2}+x+1=0 $$

Solve the equation for the indicated variable. $$ \frac{1}{s+a}+\frac{1}{s+b}=\frac{1}{c} ; \quad \text { for } s $$

\(61-70\) . Find all solutions, real and complex, of the equation. $$ x^{3}+x^{2}+x=0 $$

The given equation involves a power of the variable. Find all real solutions of the equation. \(X^{4}-16=0\)

Find all solutions of the equation, and express them in the form \(a+b i\) $$ x^{2}-3 x+3=0 $$

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ \frac{x}{x+1}>3 x $$

Solve the equation for the indicated variable. $$ \frac{1}{r}+\frac{2}{1-r}=\frac{4}{r^{2}} ; \quad \text { for } r $$

\(61-70\) . Find all solutions, real and complex, of the equation. $$ x^{4}+x^{3}+x^{2}+x=0 $$

The given equation involves a power of the variable. Find all real solutions of the equation. \(64 x^{6}=27\)

Find all solutions of the equation, and express them in the form \(a+b i\) $$ 2 x^{2}-2 x+1=0 $$

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ 1+\frac{2}{x+1} \leq \frac{2}{x} $$

Use the discriminant to determine the number of real solutions of the equation. Do not solve the equation. $$ x^{2}-6 x+1=0 $$

\(61-70\) . Find all solutions, real and complex, of the equation. $$ x^{4}-6 x^{2}+8=0 $$

The given equation involves a power of the variable. Find all real solutions of the equation. \(x^{4}+64=0\)

Dimensions of a Structure A storage bin for corn consists of a cylindrical section made of wire mesh, surmounted by a conical tin roof, as shown in the figure. The height of the roof is one-third the height of the entire structure. If the total volume of the structure is 1400\(\pi \mathrm{ft}^{3}\) and its radius is 10 \(\mathrm{ft}\) , what is its height? \([\text { Hint: Use the volume formulas listed on }\) the inside front cover of this book.

Find all solutions of the equation, and express them in the form \(a+b i\) $$ 2 x^{2}+3=2 x $$

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ \frac{3}{x-1}-\frac{4}{x} \geq 1 $$

Use the discriminant to determine the number of real solutions of the equation. Do not solve the equation. $$ x^{2}=6 x-9 $$

\(61-70\) . Find all solutions, real and complex, of the equation. $$ x^{3}+3 x^{2}+9 x+27=0 $$

The given equation involves a power of the variable. Find all real solutions of the equation. \((x-1)^{3}+8=0\)

Find all solutions of the equation, and express them in the form \(a+b i\) $$ t+3+\frac{3}{t}=0 $$

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ \frac{6}{x-1}-\frac{6}{x} \geq 1 $$

Use the discriminant to determine the number of real solutions of the equation. Do not solve the equation. $$ x^{2}+2.20 x+1.21=0 $$

\(61-70\) . Find all solutions, real and complex, of the equation. $$ x^{6}-9 x^{3}+8=0 $$

The given equation involves a power of the variable. Find all real solutions of the equation. \((x+2)^{4}-81=0\)

Historical Research Read the biographical notes on Euclid (page 63\()\) . Pythagoras (page 261\()\) , and Archimedes (page 767 ). Choose one of these mathematicians, and find out more about him from the library or on the Internet. Write a short essay on your findings. Include both biographical information and a description of the mathematics for which he is famous.

Find all solutions of the equation, and express them in the form \(a+b i\) $$ z+4+\frac{12}{2}=0 $$

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ \frac{x}{2} \geq \frac{5}{x+1}+4 $$

Use the discriminant to determine the number of real solutions of the equation. Do not solve the equation. $$ x^{2}+2.21 x+1.21=0 $$

\(61-70\) . Find all solutions, real and complex, of the equation. $$ x^{6}+9 x^{4}-4 x^{2}-36=0 $$

The given equation involves a power of the variable. Find all real solutions of the equation. \((x+1)^{4}+16=0\)

Find all solutions of the equation, and express them in the form \(a+b i\) $$ 6 x^{2}+12 x+7=0 $$

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ \frac{x+2}{x+3}<\frac{x-1}{x-2} $$

Use the discriminant to determine the number of real solutions of the equation. Do not solve the equation. $$ 4 x^{2}+5 x+\frac{13}{8}=0 $$

\(61-70\) . Find all solutions, real and complex, of the equation. $$ \sqrt{x^{2}+1}+\frac{8}{\sqrt{x^{2}+1}}=\sqrt{x^{2}+9} $$

The given equation involves a power of the variable. Find all real solutions of the equation. \(3(x-3)^{3}=375\)

Find all solutions of the equation, and express them in the form \(a+b i\) $$ 4 x^{2}-16 x+19=0 $$

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ \frac{1}{x+1}+\frac{1}{x+2} \leq 0 $$

Use the discriminant to determine the number of real solutions of the equation. Do not solve the equation. $$ 9 x^{2}-4 x+\frac{4}{9}=0 $$