# How do you factor and solve 16x^2-48x= -27?

May 23, 2015

$y = 16 {x}^{2} - 48 x + 27$ = (x - p)(x - q). I use the new AC Method (Google, Yahoo Search) to factor trinomials.
Converted trinomial: y' = x^2 - 48x + 432 = (x - p')(x - q').
Find p' and q' by composing factor pairs of a.c = 432. Proceed: ...(8, 54)(12, 36). This last sum is 12 + 36 = 48 = -b. Then p' = -12 and q' = -36. Then, $p = \frac{p '}{a} = - \frac{12}{16} = - \frac{3}{4}$ and $q = \frac{q '}{a} = - \frac{36}{16} = - \frac{9}{4.}$

Finally: $y = \left(x - \frac{3}{4}\right) \left(x - \frac{9}{4}\right) = \left(4 x - 3\right) \left(4 x - 9\right)$

Check by developing: y = 16x^2 - 36x - 12x + 27. OK

Solving y = 0 --> (4x - 3) = 0 --> $x = \frac{3}{4}$
(4x - 9) = 0 ->$x = \frac{9}{4}$

The new AC Method is fast, systematic, no guessing, no factoring by grouping and no solving binomials.