# How do you use the quadratic formula is used to find the roots of the equation x^2-6x-19=0?

May 3, 2017

${x}_{1} = 3 - 2 \sqrt{7}$

${x}_{2} = 3 + 2 \sqrt{7}$

#### Explanation:

${x}^{2} - 6 x - 19 = 0$

$\text{let us find out discriminant of the function } {x}^{2} - 6 x - 19$

$\Delta = \sqrt{{b}^{2} - 4 a c}$

$a = 1 \text{ , "b=-6" , } c = - 19$

Delta=sqrt((-6)^2-4*1*(-19)

$\Delta = \sqrt{36 + 76}$

$\Delta = \sqrt{112}$

$\Delta = \pm 4 \sqrt{7}$

${x}_{1} = \frac{- b - \Delta}{2 a} = \frac{6 - 4 \sqrt{7}}{2 \cdot 1} = \frac{6 - 4 \sqrt{7}}{2} = 3 - 2 \sqrt{7}$

${x}_{2} = \frac{- b + \Delta}{2 a} = \frac{6 + 4 \sqrt{7}}{2 \cdot 1} = \frac{6 + 4 \sqrt{7}}{2} = 3 + 2 \sqrt{7}$

May 3, 2017

Solution : $x = 3 + 2 \sqrt{7} , x = 3 - 2 \sqrt{7}$

#### Explanation:

Comparing with general equation $a {x}^{2} + b x + c = 0$

x^2-6x-19=0 ; a=1 , b= -6 , c=-19 . Discriminant $D = {b}^{2} - 4 \cdot a \cdot c = 36 + 76 = 112$ is positive so roots are real.

Quadratic formula for finding roots is$x = - \frac{b}{2 a} \pm \frac{\sqrt{{b}^{2} - 4 a c}}{2 a} \mathmr{and} x = - \frac{- 6}{2} \pm \frac{\sqrt{112}}{2} \mathmr{and} x = 3 \pm 4 \cdot \frac{\sqrt{7}}{2} \mathmr{and} x = 3 \pm 2 \cdot \sqrt{7}$

Solution : $x = 3 + 2 \sqrt{7} , x = 3 - 2 \sqrt{7}$ [Ans]