# How do you solve using the completing the square method 3x^2 + 11x – 20 = 0?

May 2, 2016

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

#### Explanation:

Premultiply by $12 = 3 \cdot {2}^{2}$ to reduce the need to do arithmetic with fractions, complete the square then use the difference of squares identity:

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

with $a = \left(6 x + 11\right)$ and $b = 19$ as follows:

$0 = 12 \left(3 {x}^{2} + 11 x - 20\right)$

$= 36 {x}^{2} + 132 x - 240$

$= {\left(6 x\right)}^{2} + 2 \left(6 x\right) \left(11\right) - 240$

$= {\left(6 x + 11\right)}^{2} - 121 - 240$

$= {\left(6 x + 11\right)}^{2} - 361$

$= {\left(6 x + 11\right)}^{2} - {19}^{2}$

$= \left(\left(6 x + 11\right) - 19\right) \left(\left(6 x + 11\right) + 19\right)$

$= \left(6 x - 8\right) \left(6 x + 30\right)$

$= \left(2 \left(3 x - 4\right)\right) \left(6 \left(x + 5\right)\right)$

$= 12 \left(3 x - 4\right) \left(x + 5\right)$

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