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How do you factor $5 x^{2} - 16 x + 12$ $5x^2-16x+12$ ?

1 Answer

Jun 27, 2015

$5 x^{2} - 16 x + 12 = (5 x - 6) ((x - 2)$ $5x^2-16x+12 = (5x-6)((x-2)$

Explanation:

Hoping for rational factors, we note that
the only factors of $5$ $5$ are $(1, 5)$ $(1,5)$

So we are looking for values $a$ $a$ and $b$ $b$
$XXXX$ $color(white)("XXXX")$ $(1 x - a) (5 x - b) = 5 x^{2} - (5 a + b) x + a b$ $(1x-a)(5x-b) = 5x^2 - (5a+b)x +ab$
such that $a \cdot b = 12$ $a*b = 12$
and $5 a + b = 16$ $5a+b = 16$

The $(a, b)$ $(a,b)$ factors of 12 are $(1, 12), (2, 6), (3, 4), (4, 3), (6, 2), (12, 1)$ $(1,12), (2,6), (3,4), (4,3), (6,2), (12,1)$

We can check each pair for a pair that satisfies $5 a + b = 16$ $5a+b = 16$
until we find $(a, b) = (2, 6)$ $(a,b)=(2,6)$
which implies our factors are
$XXXX$ $color(white)("XXXX")$ $(x - 2) (5 x - 6)$ $(x-2)(5x-6)$

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