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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 11: Q. 192 (page 1069)

In the following exercises, solve.
$15 x^{- 2} - 4 x^{- 1} - 4 = 0$

Short Answer

Expert verified

The solutions of the given equation are $x = - \frac{5}{2}$ and $x = \frac{3}{2}$ .

Step by step solution

01

Step 1. Write the equation in standard form.

The given equation is:

$15 x^{- 2} - 4 x^{- 1} - 4 = 0$

It can be written as:

$15 (x^{- 1})^{2} - 4 x^{- 1} - 4 = 0$

Substitute $x^{- 1} = u$ .

role="math" localid="1644992521952" $15 u^{2} - 4 u - 4 = 0$

02

Step 2. Solve the equation.

Split the middle term.

$\begin{array}{rcl} 15 u^{2} - 10 u + 6 u - 4 & = & 0 \\ 5 u (3 u - 2) + 2 (3 u - 2) & = & 0 \\ (5 u + 2) (3 u - 2) & = & 0 \\ u & = & - \frac{2}{5} and u = \frac{2}{3} \end{array}$

Substitute $u = x^{- 1}$ .

$\begin{array}{rcl} x^{- 1} & = & - \frac{2}{5} and x^{- 1} = \frac{2}{3} \\ x & = & - \frac{5}{2} and x = \frac{3}{2} \end{array}$

03

Step 3. Conclusion.

The solutions of the given equation are $- \frac{5}{2}$ and $x = \frac{3}{2}$ .

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