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How do you divide \frac { 6p ^ { 2} + p - 12} { 8p ^ { 2} + 18p + 9} \div \frac { 6p ^ { 2} - 11p + 4} { 2p ^ { 2} + 13p - 7}?

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Explanation

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Explanation:

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5
Apr 11, 2018

$\frac{p - 7}{4 p + 3}$

Explanation:

Let's begin by factorizing each quadratic trinomial.

=>6p^2+p−12
=6p^2+9p-8p−12
$= 3 p \left(2 p + 3\right) - 4 \left(2 p + 3\right)$
$= \textcolor{red}{\left(3 p - 4\right) \left(2 p + 3\right)}$

$\implies 8 {p}^{2} + 18 p + 9$
$= 8 {p}^{2} + 6 p + 12 p + 9$
$= 2 p \left(4 p + 3\right) + 3 \left(4 p + 3\right)$
$= \textcolor{b l u e}{\left(4 p + 3\right) \left(2 p + 3\right)}$

=>6p^2−11p+4
=6p^2−3p-8p+4
$= 3 p \left(2 p - 1\right) - 4 \left(2 p - 1\right)$
$= \textcolor{g r e e n}{\left(3 p - 4\right) \left(2 p - 1\right)}$

=>2p^2+13p−7
$= 2 {p}^{2} - 1 p + 14 p - 7$
$= p \left(2 p - 1\right) - 7 \left(2 p - 1\right)$
$= \textcolor{m a \ge n t a}{\left(p - 7\right) \left(2 p - 1\right)}$

now let's combine everything.
the question is,
color(red)(6p^2+p−12)/color(blue)(8p^2+18p+9) -: color(green)(6p^2−11p+4)/color(magenta)(2p^2+13p−7)

=> color(red)(6p^2+p−12)/color(blue)(8p^2+18p+9) xx color(magenta)(2p^2+13p−7)/color(green)(6p^2−11p+4)

$\implies \frac{\textcolor{red}{\cancel{\left(3 p - 4\right)} \cancel{\left(2 p + 3\right)}}}{\textcolor{b l u e}{\left(4 p + 3\right) \cancel{\left(2 p + 3\right)}}} \times \frac{\textcolor{m a \ge n t a}{\left(p - 7\right) \cancel{\left(2 p - 1\right)}}}{\textcolor{g r e e n}{\cancel{\left(3 p - 4\right)} \cancel{\left(2 p - 1\right)}}}$

$\implies \frac{p - 7}{4 p + 3}$

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