Chapter 1

\(61-70\) . Find all solutions, real and complex, of the equation. $$ 1-\sqrt{x^{2}+7}=6-x^{2} $$

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

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

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

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

\(71-74\) . Solve the equation for the variable \(x\) . The constants \(a\) and \(b\) represent positive real numbers. $$ x^{4}+5 a x^{2}+4 a^{2}=0 $$

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

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

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ \frac{(2 x-1)(x-3)^{2}}{x-4}<0 $$

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

\(71-74\) . Solve the equation for the variable \(x\) . The constants \(a\) and \(b\) represent positive real numbers. $$ a^{3} x^{3}+b^{3}=0 $$

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

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

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ X^{4}>X^{2} $$

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

\(71-74\) . Solve the equation for the variable \(x\) . The constants \(a\) and \(b\) represent positive real numbers. $$ \sqrt{x+a}+\sqrt{x-a}=\sqrt{2} \sqrt{x+6} $$

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

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

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ x^{5}>x^{2} $$

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

\(71-74\) . Solve the equation for the variable \(x\) . The constants \(a\) and \(b\) represent positive real numbers. $$ \sqrt{x}+a \sqrt[3]{x}+b \sqrt[6]{x}+a b=0 $$

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

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

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

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?

Find the solution of the equation rounded to two decimals. \(3.02 x+1.48=10.92\)

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

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

Solve the equation for \(x\) $$ b x^{2}+2 x+\frac{1}{b}=0 \quad(b \neq 0) $$

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?

Find the solution of the equation rounded to two decimals. \(8.36-0.95 x=9.97\)

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

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

Find all values of \(k\) that ensure that the given equation has exactly one solution. $$ 4 x^{2}+k x+25=0 $$

Fish Population A large pond is stocked with fish. The fish population \(P\) is modeled by the formula \(P=3 t+10 \sqrt{t}+140,\) where \(t\) is the number of days since the fish were first introduced into the pond. How many days will it take for the fish population to reach 500\(?\)

Find the solution of the equation rounded to two decimals. \(2.15 x-4.63=x+1.19\)

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

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

The Lens Equation If \(F\) is the focal length of a convex lens and an object is placed at a distance \(x\) from the lens, then its image will be at a distance \(y\) from the lens, where \(F, x,\) and \(y\) are related by the lens equation $$\frac{1}{F}=\frac{1}{x}+\frac{1}{y}$$ Suppose that a lens has a focal length of 4.8 \(\mathrm{cm}\) and that the image of an object is 4 \(\mathrm{cm}\) closer to the lens than the object itself. How from the lens is the object?

Find the solution of the equation rounded to two decimals. \(3.95-x=2.32 x+2.00\)

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

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

Number Problem Find two numbers whose sum is 55 and whose product is \(684 .\)

Volume of Grain Grain is falling from a chute onto the ground, forming a conical pile whose diameter is always three times its height. How high is the pile (to the nearest hundredth of a foot) when it contains 1000 \(\mathrm{ft}^{3}\) of grain?

Find the solution of the equation rounded to two decimals. \(3.16(x+4.63)=4.19(x-7.24)\)

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

Number Problem The sum of the squares of two consecutive even integers is \(1252 .\) Find the integers.

Radius of a Tank A spherical tank has a capacity of 750 gallons. Using the fact that 1 gallon is about 0.1337 \(\mathrm{ft}^{3}\) , find the radius of the tank (to the nearest hundredth of a foot).

Find the solution of the equation rounded to two decimals. \(2.14(x-4.06)=2.27-0.11 x\)

Complex Conjugate Roots Suppose that the equation \(a x^{2}+b x+c=0\) has real coefficients and complex roots. Why must the roots be complex conjugates of each other? (Think about how you would find the roots using the Quadratic Formula).