How do you factor $8 n^{2} - 9 n p - 14 p^{2}$ $8n^2-9np-14p^2$ ?

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How do you factor $8 n^{2} - 9 n p - 14 p^{2}$ $8n^2-9np-14p^2$ ?

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Answer 1 · 2016-08-28T19:37:15

Use an AC method to find:

$8 n^{2} - 9 n p - 14 p^{2} = (8 n + 7 p) (n - 2 p)$ $8n^2-9np-14p^2=(8n+7p)(n-2p)$

Explanation:

Given:

$8 n^{2} - 9 n p - 14 p^{2}$ $8n^2-9np-14p^2$

Use an AC method:

Look for a pair of factors of $A C = 8 \cdot 14 = 112$ $AC=8*14=112$ which differ by $B = 9$ $B=9$ .

Note that in order for the difference of the factors to be odd, one of the factors must be even and the other odd. So we can eliminate most possibilities quickly and find...

The pair $16, 7$ $16, 7$ works.

Use this pair to split the middle term, then factor by grouping as follows:

$8 n^{2} - 9 n p - 14 p^{2}$ $8n^2-9np-14p^2$

$= 8 n^{2} - 16 n p + 7 n p - 14 p^{2}$ $=8n^2-16np+7np-14p^2$

$= (8 n^{2} - 16 n p) + (7 n p - 14 p^{2})$ $=(8n^2-16np)+(7np-14p^2)$

$= 8 n (n - 2 p) + 7 p (n - 2 p)$ $=8n(n-2p)+7p(n-2p)$

$= (8 n + 7 p) (n - 2 p)$ $=(8n+7p)(n-2p)$

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How do you factor $8 n^{2} - 9 n p - 14 p^{2}$ $8n^2-9np-14p^2$ ?

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How do you factor $8 n^{2} - 9 n p - 14 p^{2}$ $8n^2-9np-14p^2$ ?