# How do you solve 9x ^ { 2} - 36x =6?

Jun 12, 2017

#### Answer:

$x = \frac{36 \pm 6 \sqrt{42}}{18}$

#### Explanation:

$9 {x}^{2} - 36 x = 6$

We can move $6$ to the other side, then use the quadratic formula.

$9 {x}^{2} - 36 x - 6 = 0$

$a {x}^{2} + b x + c = 0$

$a = 9$
$b = - 36$
$c = - 6$

$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

$x = \frac{- - 36 \pm \sqrt{{\left(- 36\right)}^{2} - 4 \times 9 \times - 6}}{2 \times 9}$

Now we can simplify and solve.

$x = \frac{\cancel{- -} 36 \pm \sqrt{- {36}^{2} - 4 \times 9 \times - 6}}{18}$

$x = \frac{36 \pm \sqrt{{\left(- 36\right)}^{2} - 36 \times - 6}}{18}$

$x = \frac{36 \pm \sqrt{{\left(- 36\right)}^{2} + 216}}{18}$

$x = \frac{36 \pm \sqrt{1296 + 216}}{18}$

$x = \frac{36 \pm \sqrt{1512}}{18}$

color(blue)( x =(36 +- 6sqrt42) / (18)

$\therefore$ $x = \frac{36 \pm \sqrt{1512}}{18}$

Jun 12, 2017

#### Answer:

$x = \frac{6 \pm \sqrt{42}}{3}$

#### Explanation:

$y = 9 {x}^{2} - 36 x - 6 = 0$
$y = 3 \left(3 {x}^{2} - 12 x - 2\right) = 0$
Use the improved quadratic formula (Google, Socratic Search):
$D = {d}^{2} = {b}^{2} - 4 a c = 144 + 24 = 168$ --> $d = \pm 2 \sqrt{42}$
There are 2 real roots:
$x = - \frac{b}{2 a} \pm \frac{d}{2 a} = \frac{12}{6} \pm \frac{2 \sqrt{42}}{6} = 2 \pm \frac{\sqrt{42}}{3} =$
$x = \frac{6 \pm \sqrt{42}}{3}$