Solve Using the Quadratic Formula 7x^2-7x-1=0

$7 x^{2} - 7 x - 1 = 0$

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 $a = 7$ , $b = - 7$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{7 \pm \sqrt{{(- 7)}^{2} - 4 \cdot (7 \cdot - 1)}}{2 \cdot 7}$

Step 3

Simplify.

Simplify the numerator.

Raise $- 7$ to the power of $2$ .

$x = \frac{7 \pm \sqrt{49 - 4 \cdot 7 \cdot - 1}}{2 \cdot 7}$

Multiply $- 4 \cdot 7 \cdot - 1$ .

Multiply $- 4$ by $7$ .

$x = \frac{7 \pm \sqrt{49 - 28 \cdot - 1}}{2 \cdot 7}$

Multiply $- 28$ by $- 1$ .

$x = \frac{7 \pm \sqrt{49 + 28}}{2 \cdot 7}$

Add $49$ and $28$ .

$x = \frac{7 \pm \sqrt{77}}{2 \cdot 7}$

Multiply $2$ by $7$ .

$x = \frac{7 \pm \sqrt{77}}{14}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{7 \pm \sqrt{77}}{14}$

Decimal Form:

$x = 1.12678317 \dots, - 0.12678317 \dots$