Solve Using the Quadratic Formula x^2-6x-19=0

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

Step 1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2

Substitute the values $a = 1$ , $b = - 6$ , and $c = - 19$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (1 \cdot - 19)}}{2 \cdot 1}$

Step 3

Simplify.

Simplify the numerator.

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 19}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 19$ .

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot - 19}}{2 \cdot 1}$

Multiply $- 4$ by $- 19$ .

$x = \frac{6 \pm \sqrt{36 + 76}}{2 \cdot 1}$

Add $36$ and $76$ .

$x = \frac{6 \pm \sqrt{112}}{2 \cdot 1}$

Rewrite $112$ as $4^{2} \cdot 7$ .

Factor $16$ out of $112$ .

$x = \frac{6 \pm \sqrt{16 (7)}}{2 \cdot 1}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{6 \pm \sqrt{4^{2} \cdot 7}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{6 \pm 4 \sqrt{7}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{6 \pm 4 \sqrt{7}}{2}$

Simplify $\frac{6 \pm 4 \sqrt{7}}{2}$ .

$x = 3 \pm 2 \sqrt{7}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 3 \pm 2 \sqrt{7}$

Decimal Form:

$x = 8.29150262 \dots, - 2.29150262 \dots$