# How do you solve -3x^2+7x= -5 by completing the square?

Jun 24, 2018

$x = \frac{7 \pm \sqrt{109}}{6}$

#### Explanation:

You should always start by what you want to reach, namely an expression on the form of
${\left(x - a\right)}^{2} = b$
But first, to get some idea, let's start with a graph of the function
$f \left(x\right) = - 3 {x}^{2} + 7 x + 5$:

We want ${\left(x - a\right)}^{2} = {x}^{2} - 2 a x + {a}^{2}$ on the left side
Our expression has
$- 3 {x}^{x} + 7 x = - 5$
Let's simplify by dividing the whole on $- 3$:
${x}^{2} - \frac{7}{3} x = \frac{5}{3}$

Therefore $- 2 a x = - \frac{7}{3} x$ or $2 a = \frac{7}{3}$, $a = \frac{7}{6}$
Add ${a}^{2} = {\left(\frac{7}{6}\right)}^{2}$ on both sides:
${x}^{2} - \frac{7}{3} x + {\left(\frac{7}{6}\right)}^{2} = \frac{5}{3} + \frac{49}{6} ^ 2$
${\left(x - \frac{7}{6}\right)}^{2} = \frac{5 \cdot 12 + 49}{6} ^ 2 = \frac{109}{6} ^ 2$
$x - \frac{7}{6} = \pm \frac{\sqrt{109}}{6}$
$x = \frac{7 \pm \sqrt{109}}{6}$