# How do you solve x^2 - 12x + 80 = 8?

Aug 14, 2015

$x = 6 \pm 6 i$
$\textcolor{w h i t e}{\text{XXXX}}$(there are no Real solution values)

#### Explanation:

Given ${x}^{2} - 12 x + 80 = 8$

Subtracting $8$ from both sides
$\textcolor{w h i t e}{\text{XXXX}}$${x}^{2} - 12 x + 72 = 0$

Using the quadratic formula (see below if you are uncertain of this)
$\textcolor{w h i t e}{\text{XXXX}}$x= (-(-12)+-sqrt((-12)^2-4(1)(72)))/(2(1)

$\textcolor{w h i t e}{\text{XXXXXXXX}}$=(12 +-sqrt(144 - 288)/2

$\textcolor{w h i t e}{\text{XXXXXXXX}}$$= \frac{12 \pm \sqrt{- 144}}{2}$

$\textcolor{w h i t e}{\text{XXXXXXXX}}$$= \frac{12 \pm 12 i}{2}$

$\textcolor{w h i t e}{\text{XXXXXXXX}}$$= 6 \pm 6 i$

Given the general quadratic equation in the form:
$\textcolor{w h i t e}{\text{XXXX}}$$a {x}^{2} + b x + c = 0$
the solutions are given by the quadratic formula:
$\textcolor{w h i t e}{\text{XXXX}}$$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

$\textcolor{w h i t e}{\text{XXXXXX}}$The need to use this formula comes up often enough that it is worth memorizing

Aug 14, 2015

color(blue)(x=6(1+i)
color(blue)( x=6(1-i)

#### Explanation:

x^2−12x+80=8
x^2−12x+80 -8=0

x^2−12x+72=0 .

The equation is of the form color(blue)(ax^2+bx+c=0 where:
$a = 1 , b = - 12 , c = 72$

The Discriminant is given by:
$\Delta = {b}^{2} - 4 \cdot a \cdot c$

$= {\left(- 12\right)}^{2} - \left(4 \cdot 1 \cdot 72\right)$
$= 144 - 288 = - 144$

The solutions are found using the formula
$x = \frac{- b \pm \sqrt{\Delta}}{2 \cdot a}$

$x = \frac{- \left(- 12\right) \pm \sqrt{- 144}}{2 \cdot 1} = \frac{12 \pm \sqrt{- 144}}{2}$

$x = \frac{12 \pm 12 i}{2}$

$= \frac{2 \left(6 \pm 6 i\right)}{2}$

$= 6 \pm 6 i$

color(blue)(x=6(1+i)
color(blue)( x=6(1-i)