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Mid-Chapter Mixed Review

Determine whether the statement is true or false.

The product of a complex number and its conjugate is a real number. [3.1]
Every quadratic equation has at least one x-intercept. [3.2]
If a quadratic equation has two different real-number solutions, then its discriminant is positive. [3.2]
The vertex of the graph of the function $f (x) = 3 {(x + 4)}^{2} + 5$ $f (x) = 3 {(x + 4)}^{2} + 5$ is (4, 5). [3.3]

Express the number in terms of i. [3.1]

$\sqrt{- 36}$ $\sqrt{- 36}$
$\sqrt{- 5}$ $\sqrt{- 5}$
$- \sqrt{- 16}$ $- \sqrt{- 16}$
$\sqrt{- 32}$ $\sqrt{- 32}$

Simplify. Write answers in the form $a + b i$ $a + b i$ , where a and b are real numbers. [3.1]

$(3 - 2 i) + (- 4 + 3 i)$ $(3 - 2 i) + (- 4 + 3 i)$
$(- 5 + i) - (2 - 4 i)$ $(- 5 + i) - (2 - 4 i)$
$(2 + 3 i) (4 - 5 i)$ $(2 + 3 i) (4 - 5 i)$
$\frac{3 + i}{- 2 + 5 i}$ $\frac{3 + i}{- 2 + 5 i}$

Simplify. [3.1]

$i^{13}$ $i^{13}$
$i^{44}$ $i^{44}$
${(- i)}^{5}$ ${(- i)}^{5}$
${(2 i)}^{6}$ ${(2 i)}^{6}$

Solve. [3.2]

$x^{2} + 3 x - 4 = 0$ $x^{2} + 3 x - 4 = 0$
$2 x^{2} + 6 = - 7 x$ $2 x^{2} + 6 = - 7 x$
$4 x^{2} = 24$ $4 x^{2} = 24$
$x^{2} + 100 = 0$ $x^{2} + 100 = 0$
Find the zeros of $f (x) = 4 x^{2} - 8 x - 3$ $f (x) = 4 x^{2} - 8 x - 3$ by completing the square. Show your work. [3.2]

In Exercises 22–24, (a) find the discriminant $b^{2} - 4 a c$ $b^{2} - 4 a c$ , and then determine whether one real-number solution, two different real-number solutions, or two different imaginary-number solutions exist; and (b) solve the equation, finding exact solutions and approximate solutions rounded to three decimal places, where appropriate. [3.2]

$x^{2} - 3 x - 5 = 0$ $x^{2} - 3 x - 5 = 0$
$4 x^{2} - 12 x + 9 = 0$ $4 x^{2} - 12 x + 9 = 0$
$3 x^{2} + 2 x = - 1$ $3 x^{2} + 2 x = - 1$

Solve. [3.2]

$x^{4} + 5 x^{2} - 6 = 0$ $x^{4} + 5 x^{2} - 6 = 0$
$2 x - 5 \sqrt{x} + 2 = 0$ $2 x - 5 \sqrt{x} + 2 = 0$
One number is 2 more than another. The product of the numbers is 35. Find the numbers. [3.2]

In Exercises 28 and 29, (a) find the vertex; (b) find the axis of symmetry; (c) determine whether there is a maximum or a minimum value, and find that value; (d) find the range; (e) find the intervals on which the function is increasing or decreasing; and (f ) graph the function. [3.3]

$f (x) = x^{2} - 6 x + 7$ $f (x) = x^{2} - 6 x + 7$
$f (x) = - 2 x^{2} - 4 x - 5$ $f (x) = - 2 x^{2} - 4 x - 5$
The sum of the base and the height of a triangle is 16 in. Find the dimensions for which the area is a maximum. [3.3]

Collaborative Discussion and Writing

Is the sum of two imaginary numbers always an imaginary number? Explain your answer. [3.1]
The graph of a quadratic function can have 0, 1, or 2 x-intercepts. How can you predict the number of x-intercepts without drawing the graph or (completely) solving an equation? [3.2]
Discuss two ways in which we used completing the square in this chapter. [3.2], [3.3]
Suppose that the graph of $f (x) = a x^{2} + b x + c$ $f (x) = a x^{2} + b x + c$ has x-intercepts $(x_{1}, 0)$ $(x_{1}, 0)$ and $(x_{2}, 0)$ $(x_{2}, 0)$ . What are the x-intercepts of $g (x) = - a x^{2} - b x - c$ $g (x) = - a x^{2} - b x - c$ ? Explain. [3.3]

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Table of Contents for Mid-Chapter Mixed Review

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Table of Contents for
Mid-Chapter Mixed Review