# How do you solve using the completing the square method x^2+24x+90=0?

Apr 29, 2016

${\left(x + 12\right)}^{2} - 54 = 0$

#### Explanation:

The expression ${\left(x + a\right)}^{2}$ expands as ${x}^{2} + 2 a x + {a}^{2}$
so to complete the square we use half the coefficient of the middle term to be $a$.

We then subtract the equivalent of ${a}^{2}$ (in this case ${12}^{2}$) and add the final term ($90$).

${\left(x + 12\right)}^{2} - 144 + 90$
$= {\left(x + 12\right)}^{2} - 54$

Apr 29, 2016

$\implies x = - 12 \pm 3 \sqrt{6} \text{ }$ as exact values

$\implies x \approx - 4.65 \text{ and "-19.35" }$ to 2 decimal places

#### Explanation:

Standard for $\text{ "y=ax^2+bx+c}$

Write as$\text{ } y = a {\left(x + \frac{b}{2 a}\right)}^{2} + c + \left(- {b}^{2} / \left(4 a\right)\right)$

The purpose of the ${b}^{2} / \left(4 a\right)$ is to mathematically remove an error that have been introduced by building $a {\left(x + \frac{b}{2 a}\right)}^{2}$

If you square $\frac{b}{2 a}$ then multiply it out by the variable $' a '$ in front of the bracket you have introduced a value that was not in the original equation. So you remove it by subtraction.
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Given:$\text{ "x^2+24x+90=0" }$ Note that $a = 1$

Write as:$\text{ } y = {\left(x + 12\right)}^{2} + 90 + k$

But $k = - {\left(12\right)}^{2} / 4 = - 144$

$\textcolor{b r o w n}{\text{ "=>y=(x+12)^2+90+k)color(blue)(" "->" } y = {\left(x + 12\right)}^{2} - 54}$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

${x}^{2} + 24 x + 90 = y = 0 = {\left(x + 12\right)}^{2} - 54$

So ${\left(x + 12\right)}^{2} = + 54$

$\implies \sqrt{{\left(x + 12\right)}^{2}} = \sqrt{54}$

$\implies x + 12 = \pm \sqrt{6 \times {3}^{2}}$

$\implies x = - 12 \pm 3 \sqrt{6}$

$\implies x \approx - 4.65 \text{ and } - 19.35$ to 2 decimal places