Chapter 3

These exercises review topics covered in earlier sections. The concepts are used in solving the applications that follow in this exercise set. Do not use a calculator. Find the maximum y-value on the graph of \(y=-16 x^{2}+32 x+100\).

For each complex number, (a) state the real part, (b) state the imaginary part, and (c) identify the number as one or more of the following: real, pure imaginary, or nonreal complex. $$-9 i$$

These exercises review topics covered in earlier sections. The concepts are used in solving the applications that follow in this exercise set. Do not use a calculator. Find the maximum y-value on the graph of \(y=-2 x^{2}+8 x-5\).

For each complex number, (a) state the real part, (b) state the imaginary part, and (c) identify the number as one or more of the following: real, pure imaginary, or nonreal complex. $$3 i$$

These exercises review topics covered in earlier sections. The concepts are used in solving the applications that follow in this exercise set. Do not use a calculator. Find the minimum function value of \(f(x)=3 x^{2}-24 x+50\).

Match each equation in Column I with its solution \((s)\) in Column \(I I\). A. \(\pm 2 i\) B. \(\pm 2 \sqrt{2}\) C. \(\pm i \sqrt{2}\) D. 2 E. \(\pm \sqrt{2} \quad\) F. \(-2\) G. \(\pm 2\) H. \(\pm 2 i \sqrt{2}\) $$x^{2}+2=0$$

For each complex number, (a) state the real part, (b) state the imaginary part, and (c) identify the number as one or more of the following: real, pure imaginary, or nonreal complex. $$\pi$$

These exercises review topics covered in earlier sections. The concepts are used in solving the applications that follow in this exercise set. Do not use a calculator. Find the minimum function value of \(f(x)=5 x^{2}+30 x+17\).

Match each equation in Column I with its solution \((s)\) in Column \(I I\). A. \(\pm 2 i\) B. \(\pm 2 \sqrt{2}\) C. \(\pm i \sqrt{2}\) D. 2 E. \(\pm \sqrt{2} \quad\) F. \(-2\) G. \(\pm 2\) H. \(\pm 2 i \sqrt{2}\) $$x^{2}-2=0$$

For each complex number, (a) state the real part, (b) state the imaginary part, and (c) identify the number as one or more of the following: real, pure imaginary, or nonreal complex. $$3 i$$

These exercises review topics covered in earlier sections. The concepts are used in solving the applications that follow in this exercise set. Do not use a calculator. Solve \(-4 x^{2}+5 x=1\).

Match each equation in Column I with its solution \((s)\) in Column \(I I\). A. \(\pm 2 i\) B. \(\pm 2 \sqrt{2}\) C. \(\pm i \sqrt{2}\) D. 2 E. \(\pm \sqrt{2} \quad\) F. \(-2\) G. \(\pm 2\) H. \(\pm 2 i \sqrt{2}\) $$x^{2}=-8$$

For each quadratic function, (a) write the function in the form \(P(x)=a(x-h)^{2}+k,\) (b) give the vertex of the parabola, and (c) graph the function. Do not use a calculator. $$P(x)=x^{2}-2 x-15$$

For each complex number, (a) state the real part, (b) state the imaginary part, and (c) identify the number as one or more of the following: real, pure imaginary, or nonreal complex. $$3+7 i$$

These exercises review topics covered in earlier sections. The concepts are used in solving the applications that follow in this exercise set. Do not use a calculator. Solve \(x^{2}-6 x=7\).

For each quadratic function, (a) write the function in the form \(P(x)=a(x-h)^{2}+k,\) (b) give the vertex of the parabola, and (c) graph the function. Do not use a calculator. $$P(x)=x^{2}+2 x-15$$

For each complex number, (a) state the real part, (b) state the imaginary part, and (c) identify the number as one or more of the following: real, pure imaginary, or nonreal complex. $$-8+4 i$$

These exercises review topics covered in earlier sections. The concepts are used in solving the applications that follow in this exercise set. Do not use a calculator. Solve \(\frac{1}{2} x^{2}+3=6 x\).

Match each equation in Column I with its solution \((s)\) in Column \(I I\). A. \(\pm 2 i\) B. \(\pm 2 \sqrt{2}\) C. \(\pm i \sqrt{2}\) D. 2 E. \(\pm \sqrt{2} \quad\) F. \(-2\) G. \(\pm 2\) H. \(\pm 2 i \sqrt{2}\) $$x-2=0$$

For each quadratic function, (a) write the function in the form \(P(x)=a(x-h)^{2}+k,\) (b) give the vertex of the parabola, and (c) graph the function. Do not use a calculator. $$P(x)=-x^{2}-3 x+10$$

For each complex number, (a) state the real part, (b) state the imaginary part, and (c) identify the number as one or more of the following: real, pure imaginary, or nonreal complex. $$i \sqrt{7}$$

These exercises review topics covered in earlier sections. The concepts are used in solving the applications that follow in this exercise set. Do not use a calculator. Solve \(\frac{1}{4} x^{2}+x=1\).

Match each equation in Column I with its solution \((s)\) in Column \(I I\). A. \(\pm 2 i\) B. \(\pm 2 \sqrt{2}\) C. \(\pm i \sqrt{2}\) D. 2 E. \(\pm \sqrt{2} \quad\) F. \(-2\) G. \(\pm 2\) H. \(\pm 2 i \sqrt{2}\) $$x+2=0$$

For each quadratic function, (a) write the function in the form \(P(x)=a(x-h)^{2}+k,\) (b) give the vertex of the parabola, and (c) graph the function. Do not use a calculator. $$P(x)=-x^{2}+3 x+10$$

For each complex number, (a) state the real part, (b) state the imaginary part, and (c) identify the number as one or more of the following: real, pure imaginary, or nonreal complex. $$-i \sqrt{3}$$

Solve each problem. For the rectangular parking area shown, which equation says that the area is \(40,000\) square yards? A. \(x(2 x+200)=40,000\) B. \(2 x+2(2 x+200)=40,000\) C. \(x+(2 x+200)=40,000\) D. None of the above

Which one of the following equations is set up for direct use of the zero- product property? Solve it. A. \(3 x^{2}-17 x-6=0\) B. \((2 x+5)^{2}=7\) C. \(x^{2}+x=12\) D. \((3 x+1)(x-7)=0\)

For each quadratic function, (a) write the function in the form \(P(x)=a(x-h)^{2}+k,\) (b) give the vertex of the parabola, and (c) graph the function. Do not use a calculator. $$P(x)=x^{2}-6 x$$

For each complex number, (a) state the real part, (b) state the imaginary part, and (c) identify the number as one or more of the following: real, pure imaginary, or nonreal complex. $$\sqrt{-7}$$

Which one of the following equations is set up for direct use of the square root property? Solve it. A. \(3 x^{2}-17 x-6=0\) B. \((2 x+5)^{2}=7\) C. \(x^{2}+x=12\) D. \((3 x+1)(x-7)=0\)

For each quadratic function, (a) write the function in the form \(P(x)=a(x-h)^{2}+k,\) (b) give the vertex of the parabola, and (c) graph the function. Do not use a calculator. $$P(x)=x^{2}+4 x$$

For each complex number, (a) state the real part, (b) state the imaginary part, and (c) identify the number as one or more of the following: real, pure imaginary, or nonreal complex. $$\sqrt{-10}$$

Solve each problem. Suppose that \(x\) represents one of two positive numbers whose sum is \(30 .\) (a) Represent the other of the two numbers in terms of \(x .\) (b) What are the restrictions on \(x ?\) (c) Determine a function \(P\) that represents the product of the two numbers. (d) Determine analytically and support graphically the two such numbers whose product is a maximum. What is this maximum product?

Only one of the following equations does not require Step 1 of the method for completing the square. Which one is it? Solve it. A. \(3 x^{2}-17 x-6=0\) B. \((2 x+5)^{2}=7\) C. \(x^{2}+x=12\) D. \((3 x+1)(x-7)=0\)

For each quadratic function, (a) write the function in the form \(P(x)=a(x-h)^{2}+k,\) (b) give the vertex of the parabola, and (c) graph the function. Do not use a calculator. $$P(x)=2 x^{2}-2 x-24$$

Perform operations and write the result in standard form. $$3 i+5 i$$

Solve each problem. Suppose that \(x\) represents one of two positive numbers whose sum is 45. (a) Represent the other of the two numbers in terms of \(x .\) (b) What are the restrictions on \(x ?\) (c) Determine a function \(P\) that represents the product of the two numbers. (d) For what two such numbers is the product equal to \(504 ?\) Determine analytically. (e) Determine analytically and support graphically the two such numbers whose product is a maximum. What is this maximum product?

Only one of the following equations is set up so that the values of \(a, b,\) and \(c\) can be determined immediately. Which one is it? Solve it. A. \(3 x^{2}-17 x-6=0\) B. \((2 x+5)^{2}=7\) C. \(x^{2}+x=12\) D. \((3 x+1)(x-7)=0\)

For each quadratic function, (a) write the function in the form \(P(x)=a(x-h)^{2}+k,\) (b) give the vertex of the parabola, and (c) graph the function. Do not use a calculator. $$P(x)=3 x^{2}+3 x-6$$

Perform operations and write the result in standard form. $$5 i-(2-i)$$

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}=16$$

Solve each problem. A farmer has 1000 feet of fence to enclose a rectangular area. What dimensions for the rectangle result in the maximum area enclosed by the fence?

For each quadratic function, (a) write the function in the form \(P(x)=a(x-h)^{2}+k,\) (b) give the vertex of the parabola, and (c) graph the function. Do not use a calculator. $$P(x)=-2 x^{2}+6 x$$

Perform operations and write the result in standard form. $$(-7 i)(1+i)$$

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}=144$$

Solve each problem. A homeowner has 80 feet of fence to enclose a rectangular garden. What dimensions for the garden give the maximum area?

For each quadratic function, (a) write the function in the form \(P(x)=a(x-h)^{2}+k,\) (b) give the vertex of the parabola, and (c) graph the function. Do not use a calculator. $$P(x)=-4 x^{2}+4 x$$

Perform operations and write the result in standard form. $$\frac{4+2 i}{i}$$

Solve each equation. For equations with real solutions, support your answers graphically. $$2 x^{2}=90$$

Solve each problem. American River College has plans to construct a rectangular parking lot on land bordered on one side by a highway. There are 640 feet of fencing available to fence the other three sides. Let \(x\) represent the length of each of the two parallel sides of fencing. (a) Express the length of the remaining side to be fenced in terms of \(x\). (b) What are the restrictions on \(x ?\) (c) Determine a function \(s l\) that represents the area of the parking lot in terms of \(x\). (d) Graph the function \(\mathscr{A}\) from part (c) in a viewing window of \([0,320]\) by \([0,55,000] .\) Determine graphically the values of \(x\) that will give an area between \(30,000\) and \(40,000\) square feet. (e) What dimensions will give a maximum area, and what will this area be? Determine analytically and support graphically.