Solve Using the Quadratic Formula x^2-10x+19=0

$x^{2} - 10 x + 19 = 0$

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 $a = 1$ , $b = - 10$ , and $c = 19$ into the quadratic formula and solve for $x$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (1 \cdot 19)}}{2 \cdot 1}$

Step 3

Simplify.

Simplify the numerator.

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot 19}}{2 \cdot 1}$

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

Multiply $- 4$ by $1$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 19}}{2 \cdot 1}$

Multiply $- 4$ by $19$ .

$x = \frac{10 \pm \sqrt{100 - 76}}{2 \cdot 1}$

Subtract $76$ from $100$ .

$x = \frac{10 \pm \sqrt{24}}{2 \cdot 1}$

Rewrite $24$ as $2^{2} \cdot 6$ .

Factor $4$ out of $24$ .

$x = \frac{10 \pm \sqrt{4 (6)}}{2 \cdot 1}$

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

$x = \frac{10 \pm \sqrt{2^{2} \cdot 6}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{10 \pm 2 \sqrt{6}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{10 \pm 2 \sqrt{6}}{2}$

Simplify $\frac{10 \pm 2 \sqrt{6}}{2}$ .

$x = 5 \pm \sqrt{6}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 5 \pm \sqrt{6}$

Decimal Form:

$x = 7.44948974 \dots, 2.55051025 \dots$