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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 7: Q.494. (page 742)

Solve the equation.
$\frac{z}{2 z + 8} - \frac{3}{4 z - 8} = \frac{3 z^{2} - 16 z - 16}{8 z^{2} + 2 z - 64}$

Short Answer

Expert verified

The given equation has no solution.

Step by step solution

01

Step 1. Given information.

The given equation is $\frac{z}{2 z + 8} - \frac{3}{4 z - 8} = \frac{3 z^{2} - 16 z - 16}{8 z^{2} + 2 z - 64}$ .

02

Step 2. Simplify the equation.

Solve the given equation.

\begin{array}{rcl} \frac{z}{2 z + 8} - \frac{3}{4 z - 8} & = & \frac{3 z^{2} - 16 z - 16}{8 z^{2} + 2 z - 64} \\ \frac{z (4 z - 8) - 3 (2 z + 8)}{(2 z + 8) (4 z - 8)} & = & \frac{3 z^{2} - 16 z - 16}{8 z^{2} + 2 z - 64} \\ \frac{4 z^{2} - 8 z - 6 z - 24}{8 z^{2} - 16 z + 32 z - 64} & = & \frac{3 z^{2} - 16 z - 16}{8 z^{2} + 2 z - 64} \\ \frac{4 z^{2} - 14 z - 24}{8 z^{2} + 16 z - 64} & = & \frac{3 z^{2} - 16 z - 16}{8 z^{2} + 2 z - 64} \end{array}

03

Step 3. Continue the above calculation.

Further simplify the obtained equation.

\begin{array}{rcl} (4 z^{2} - 14 z - 24) (8 z^{2} + 2 z - 64) & = & (3 z^{2} - 16 z - 16) (8 z^{2} + 16 z - 64) \\ 32 z^{4} + 8 z^{3} - 256 z^{2} - 112 z^{3} - 28 z^{2} + 896 z - 192 z^{2} - 48 z + 1536 & = & 24 z^{4} + 48 z^{3} - 192 z^{2} - 128 z^{3} - 256 z^{2} + 1024 z - 128 z^{2} - 256 z + 1024 \\ 32 z^{4} - 104 z^{3} - 476 z^{2} + 848 z + 1536 & = & 24 z^{4} - 80 z^{3} - 576 z^{2} + 786 z + 1024 \\ 8 z^{4} - 24 z^{3} + 100 z^{2} + 80 z + 512 & = & 0 \\ 2 z^{4} - 6 z^{3} + 25 z^{2} + 20 z + 128 & = & 0 \end{array}

Using a calculator, the solution of this simplified equation can be easily found.

Hence, the given equation has no solution.

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