# How do you find all the real and complex roots of 12x²+8x-15=0?

Feb 12, 2016

The equation has two rational roots $\frac{5}{6}$ and $- \frac{3}{2}$.

#### Explanation:

Comparing the equation 12x²+8x−15=0, with general form of a quadratic equation i.e. ax²+bx+c=0, we observe that $a = 12 , b = 8 \mathmr{and} c - - 15$. Note that a quadratic equation in one variable will have two roots.

Hence discriminant ${b}^{2} - 4 a c$ equals ${8}^{2} - 4 \cdot 12 \cdot \left(- 15\right)$ or $64 + 720$ ie. $784$.

As in this equation discriminant ${b}^{2} - 4 a c \ge 0$, the roots are real and as $\sqrt{784} = 28$, roots are rational.

Roots of a general quadratic equation ax²+bx+c=0 are $\frac{- b \pm \sqrt{{b}^{2} - 4 a}}{2} a$. Hence the roots are

$\frac{- 8 \pm 28}{24}$ i.e. $\frac{5}{6}$ and $- \frac{3}{2}$.