How do you solve using the completing the square method x^2 + 12x + 20 = 0?

Mar 27, 2016

For an in depth explanation of method see
http://socratic.org/s/at7MQHg2

$\textcolor{b l u e}{x \approx - 1.101 \text{ } x \approx - 10.899}$

Explanation:

Using Shortcuts

Given:$\text{ } y = {x}^{2} + 12 x + 12$

Let $k$ be the error adjustment constant

$y = {x}^{2} + 12 x + 12 + k$

Step 1:$\text{ } y = \left({x}^{\textcolor{m a \ge n t a}{2}} + 12 x\right) + 12 + k$

Step 2:$\text{ } y = {\left(x + \textcolor{g r e e n}{12} x\right)}^{\textcolor{m a \ge n t a}{2}} + 12 + k$

Step 3:$\text{ use } \left(\frac{1}{2}\right) \times \left(\textcolor{g r e e n}{12}\right) = \textcolor{m a \ge n t a}{6}$

$\implies y = {\left(x \textcolor{m a \ge n t a}{+ 6}\right)}^{2} + 12 + k$

Step 4:$\text{ } {\textcolor{m a \ge n t a}{\left(+ 6\right)}}^{2} + k = 0$

$\implies k = \textcolor{g r e e n}{- 36}$

$\implies y = {\left(x + 6\right)}^{2} + 12 - 36$

$\implies y = {\left(x + 6\right)}^{2} - 24$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~

At $y = 0 \text{; } x = \pm \sqrt{24} - 6$

$\textcolor{b l u e}{x \approx - 1.101 \text{ } x \approx - 10.899}$

$\textcolor{m a \ge n t a}{\text{Notice that plots for both equations match}}$