Chapter 1

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

\(47-70\) The given equation involves a power of the variable. Find all real solutions of the equation. $$ x^{4}+64=0 $$

\(33-66\) . Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ \frac{x}{2} \geq \frac{5}{x+1}+4 $$

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

55–64 ? Find all solutions, real and complex, of the equation. $$ x^{6}+9 x^{4}-4 x^{2}-36=0 $$

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

\(47-70\) The given equation involves a power of the variable. Find all real solutions of the equation. $$ (x-1)^{3}+8=0 $$

\(33-66\) . Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ \frac{x+2}{x+3}<\frac{x-1}{x-2} $$

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

55–64 ? Find all solutions, real and complex, of the equation. $$ \sqrt{x^{2}+1}+\frac{8}{\sqrt{x^{2}+1}}=\sqrt{x^{2}+9} $$

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

\(47-70\) The given equation involves a power of the variable. Find all real solutions of the equation. $$ (x+2)^{4}-81=0 $$

\(33-66\) . Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ \frac{1}{x+1}+\frac{1}{x+2} \leq 0 $$

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

55–64 ? Find all solutions, real and complex, of the equation. $$ 1-\sqrt{x^{2}+7}=6-x^{2} $$

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

\(47-70\) The given equation involves a power of the variable. Find all real solutions of the equation. $$ (x+1)^{4}+16=0 $$

\(33-66\) . Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ x^{4}>x^{2} $$

Find all solutions of the equation and express them in the form \(a+b i .\) $$ \frac{1}{2} x^{2}-x+5=0 $$

65–68 ? Solve the equation for the variable x. The constants a and b represent positive real numbers. $$ x^{4}+5 a x^{2}+4 a^{2}=0 $$

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

\(47-70\) The given equation involves a power of the variable. Find all real solutions of the equation. $$ 3(x-3)^{3}=375 $$

\(33-66\) . Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ x^{5}>x^{2} $$

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

65–68 ? Solve the equation for the variable x. The constants a and b represent positive real numbers. $$ a^{3} x^{3}+b^{3}=0 $$

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

\(47-70\) The given equation involves a power of the variable. Find all real solutions of the equation. $$ 4(x+2)^{5}=1 $$

\(67-70=\) Determine the values of the variable for which the expression is defined as a real number. $$ \sqrt{16-9 x^{2}} $$

Recall that the symbol \(\overline{z}\) represents the complex con- jugate of \(z .\) If \(z=a+b i\) and \(w=c+d i,\) prove each statement. $$ \overline{z}+\overline{w}=\overline{z+w} $$

65–68 ? Solve the equation for the variable x. The constants a and b represent positive real numbers. $$ \sqrt{x+a}+\sqrt{x-a}=\sqrt{2} \sqrt{x+6} $$

Use the discriminant to determine the number of real solutions of the equation. Do not solve the equation. \(x^{2}+r x-s=0 \quad(s>0)\)

\(47-70\) The given equation involves a power of the variable. Find all real solutions of the equation. $$ \sqrt[3]{x}=5 $$

\(67-70=\) Determine the values of the variable for which the expression is defined as a real number. $$ \sqrt{3 x^{2}-5 x+2} $$

Recall that the symbol \(\overline{z}\) represents the complex con- jugate of \(z .\) If \(z=a+b i\) and \(w=c+d i,\) prove each statement. $$ \overline{z w}=\overline{z} \cdot \overline{w} $$

65–68 ? Solve the equation for the variable x. The constants a and b represent positive real numbers. $$ \sqrt{x}+a \sqrt[3]{x}+b \sqrt[6]{x}+a b=0 $$

Use the discriminant to determine the number of real solutions of the equation. Do not solve the equation. \(x^{2}-r x+s=0 \quad(s>0, r>2 \sqrt{s})\)

\(47-70\) The given equation involves a power of the variable. Find all real solutions of the equation. $$ x^{4 / 3}-16=0 $$

\(67-70=\) Determine the values of the variable for which the expression is defined as a real number. $$ \left(\frac{1}{x^{2}-5 x-14}\right)^{1 / 2} $$

Recall that the symbol \(\overline{z}\) represents the complex con- jugate of \(z .\) If \(z=a+b i\) and \(w=c+d i,\) prove each statement. $$ (\overline{z})^{2}=\overline{z^{2}} $$

Chartering a Bus A social club charters a bus at a cost of \(\$ 900\) to take a group of members on an excursion to Atlantic City. At the last minute, five people in the group decide not to go. This raises the transportation cost per person by \(\$ 2 .\) How many people originally intended to take the trip?

Solve the equation for \(x\). \(a^{2} x^{2}+2 a x+1=0 \quad(a \neq 0)\)

\(47-70\) The given equation involves a power of the variable. Find all real solutions of the equation. $$ 2 x^{5 / 3}+64=0 $$

\(67-70=\) Determine the values of the variable for which the expression is defined as a real number. $$ \sqrt[4]{\frac{1-x}{2+x}} $$

Recall that the symbol \(\overline{z}\) represents the complex con- jugate of \(z .\) If \(z=a+b i\) and \(w=c+d i,\) prove each statement. $$ \overline{\overline{z}}=z $$

Buying a Cottage A group of friends decides to buy a vacation home for \(\$ 120,000,\) sharing the cost equally. If they can find one more person to join them, each person's contribution will drop by \(\$ 6000\) . How many people are in the group?

Solve the equation for \(x\). \(b^{2} x^{2}-5 b x+4=0 \quad(b \neq 0)\)

\(47-70\) The given equation involves a power of the variable. Find all real solutions of the equation. $$ 6 x^{2 / 3}-216=0 $$

Solve the inequality for \(x,\) assuming that \(a, b,\) and \(c\) are positive constants. (a) \(a(b x-c) \geq b c \quad\) (b) \(a \leq b x+c<2 a\)

Recall that the symbol \(\overline{z}\) represents the complex con- jugate of \(z .\) If \(z=a+b i\) and \(w=c+d i,\) prove each statement. \(z+\overline{z}\) is a real number

Solve the equation for \(x\). \(a x^{2}-(2 a+1) x+(a+1)=0 \quad(a \neq 0)\)