# How do you solve the quadratic 3x^2-5x-12=0 using any method?

Jan 8, 2017

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

#### Explanation:

Method 1 - Completing the square

$0 = 12 \left(3 {x}^{2} - 5 x - 12\right)$

$\textcolor{w h i t e}{0} = 36 {x}^{2} - 60 x - 144$

$\textcolor{w h i t e}{0} = {\left(6 x\right)}^{2} - 2 \left(6 x\right) \left(5\right) + 25 - 169$

$\textcolor{w h i t e}{0} = \left(6 x - 5\right) 2 - {13}^{2}$

$\textcolor{w h i t e}{0} = \left(\left(6 x - 5\right) - 13\right) \left(\left(6 x - 5\right) + 13\right)$

$\textcolor{w h i t e}{0} = \left(6 x - 18\right) \left(6 x + 8\right)$

$\textcolor{w h i t e}{0} = \left(6 \left(x - 3\right)\right) \left(2 \left(3 x + 4\right)\right)$

$\textcolor{w h i t e}{0} = 12 \left(x - 3\right) \left(3 x + 4\right)$

Hence:

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

$\textcolor{w h i t e}{}$
Method 2 - AC method

Given:

$3 {x}^{2} - 5 x - 12$

Find a pair of factors of $A C = 3 \cdot 12 = 36$ which differ by $B = 5$

The pair $9 , 4$ works.

Use this pair to split the middle term, then factor by grouping:

$0 = 3 {x}^{2} - 5 x - 12$

$\textcolor{w h i t e}{0} = \left(3 {x}^{2} - 9 x\right) + \left(4 x - 12\right)$

$\textcolor{w h i t e}{0} = 3 x \left(x - 3\right) + 4 \left(x - 3\right)$

$\textcolor{w h i t e}{0} = \left(3 x + 4\right) \left(x - 3\right)$

Hence:

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

$\textcolor{w h i t e}{}$

The equation:

$3 {x}^{2} - 5 x - 12 = 0$

is in the form:

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

with $a = 3$, $b = - 5$, $c = - 12$

The roots are given by the quadratic formula:

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

$\textcolor{w h i t e}{x} = \frac{5 \pm \sqrt{{\left(- 5\right)}^{2} - 4 \left(3\right) \left(- 12\right)}}{2 \left(3\right)}$

$\textcolor{w h i t e}{x} = \frac{5 \pm \sqrt{25 + 144}}{6}$

$\textcolor{w h i t e}{x} = \frac{5 \pm \sqrt{169}}{6}$

$\textcolor{w h i t e}{x} = \frac{5 \pm 13}{6}$

That is:

$x = \frac{5 + 13}{6} = \frac{18}{6} = 3 \text{ }$ or $\text{ } x = \frac{5 - 13}{6} = - \frac{8}{6} = - \frac{4}{3}$