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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 9: Q 367 (page 977)

In the following exercises, 鈸� solve graphically and 鈸� write the solution in interval notation $- x^{2} - 3 x + 18 鈮� 0$

Short Answer

Expert verified

The solution in interval notation is $鈥� (- 鈭� 鈥�, 鈥� - 6] 鈭� 鈥� [3, 鈥� 鈭� 鈥�)$

Step by step solution

01

Step 1. Given information

$- x^{2} - 3 x + 18 鈮� 0$

02

Step 2. Solve graphically

The quadratic equation in standard form.

$- x^{2} - 3 x + 18 鈮� 0$

Graph the function $f (x) = - x^{2} - 3 x + 18$

The parabola opens downward.

Find the line of symmetry

$x = - \frac{b}{2 a} x = - 1.5$

Find the vertex.

$f (x) = - x^{2} - 3 x + 18 f (- 1.5) = - {(- 1.5)}^{2} - 3 (- 1.5) + 18 = 19.5$

vertex is $(- 1.5, 19.5)$

Find the x-intercepts. Let $f (x) = 0$

$f (x) = - x^{2} - 3 x + 18 0 = - (x - 3) (x + 6) x = 3 a n d x = - 6$

The x-intercepts are $(3, 0) a n d (- 6, 0)$

The graph of the parabola is

03

Step 3. Solution

The solution in interval notation is $鈥� (- 鈭� 鈥�, 鈥� - 6] 鈭� 鈥� [3, 鈥� 鈭� 鈥�)$

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