# How do you factor and solve x^2+3x+1=0?

Jun 5, 2016

#### Answer:

$x = - \frac{3}{2} + \frac{\sqrt{5}}{2}$ or $x = - \frac{3}{2} - \frac{\sqrt{5}}{2}$

#### Explanation:

Use the difference of squares identity:

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

with $a = \left(x + \frac{3}{2}\right)$ and $b = \frac{\sqrt{5}}{2}$ as follows:

$0 = {x}^{2} + 3 x + 1$

$= {\left(x + \frac{3}{2}\right)}^{2} - {3}^{2} / {2}^{2} + 1$

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

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

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

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

Hence:

$x = - \frac{3}{2} + \frac{\sqrt{5}}{2}$ or $x = - \frac{3}{2} - \frac{\sqrt{5}}{2}$