# 9/16x^2+7.5x=−25?

Mar 8, 2018

By moving the -25 to the left-hand side of the equation, one can factor the formula to get x, which is $- 6 \frac{2}{3}$

#### Explanation:

$\frac{9}{16} {x}^{2} + 7.5 x + 25 = 0$

Based on the fact that 9/16 and 25 have clean square roots (3/4 and 5, respectively) it occurred to me that the equation factored as such:

$0 = {\left(\frac{3}{4} x + 5\right)}^{2} = {\left(\frac{3}{4} x\right)}^{2} + 2 \cdot 5 \cdot \left(\frac{3}{4} x\right) + {5}^{2}$
$0 = \frac{9}{16} {x}^{2} + \frac{30}{4} x + 25$
$0 = \frac{9}{16} {x}^{2} + 7.5 x + 25$

Using the factored version, $0 = {\left(\frac{3}{4} x + 5\right)}^{2}$, we can now solve for x

$\sqrt{0} = \sqrt{{\left(\frac{3}{4} x + 5\right)}^{2}}$

$0 = \frac{3}{4} x + 5$

$- 5 = \frac{3}{4} x$

$- \frac{20}{3} = x = - 6 \frac{2}{3}$

-20$\div$3