# How do you factor 6x^2+7x-62?

Nov 19, 2016

$f \left(x\right) = 6 \left(x + \frac{7}{12} - \frac{\sqrt{1537}}{12}\right) \left(x + \frac{7}{12} + \frac{\sqrt{1537}}{12}\right)$

#### Explanation:

The difference of squares identity can be written:

${a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$

We use this with $a = \left(12 x + 7\right)$ and $b = \sqrt{1537}$ later.

Given:

$f \left(x\right) = 6 {x}^{2} + 7 x - 62$

$24 f \left(x\right) = 144 {x}^{2} + 168 x - 1488$

$\textcolor{w h i t e}{24 f \left(x\right)} = {\left(12 x + 7\right)}^{2} - 49 - 1488$

$\textcolor{w h i t e}{24 f \left(x\right)} = {\left(12 x + 7\right)}^{2} - 1537$

$\textcolor{w h i t e}{24 f \left(x\right)} = {\left(12 x + 7\right)}^{2} - {\left(\sqrt{1537}\right)}^{2}$

$\textcolor{w h i t e}{24 f \left(x\right)} = \left(\left(12 x + 7\right) - \sqrt{1537}\right) \left(\left(12 x + 7\right) + \sqrt{1537}\right)$

$\textcolor{w h i t e}{24 f \left(x\right)} = \left(12 x + 7 - \sqrt{1537}\right) \left(12 x + 7 + \sqrt{1537}\right)$

$\textcolor{w h i t e}{24 f \left(x\right)} = 144 \left(x + \frac{7}{12} - \frac{\sqrt{1537}}{12}\right) \left(x + \frac{7}{12} + \frac{\sqrt{1537}}{12}\right)$

Hence:

$f \left(x\right) = 6 \left(x + \frac{7}{12} - \frac{\sqrt{1537}}{12}\right) \left(x + \frac{7}{12} + \frac{\sqrt{1537}}{12}\right)$