Solve Using the Quadratic Formula 6x-5=-x^2

$6 x - 5 = - x^{2}$

Step 1

Add $x^{2}$ to both sides of the equation.

$6 x - 5 + x^{2} = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 1$ , $b = 6$ , and $c = - 5$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (1 \cdot - 5)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

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

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 5}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 5$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 5}}{2 \cdot 1}$

Multiply $- 4$ by $- 5$ .

$x = \frac{- 6 \pm \sqrt{36 + 20}}{2 \cdot 1}$

Add $36$ and $20$ .

$x = \frac{- 6 \pm \sqrt{56}}{2 \cdot 1}$

Rewrite $56$ as $2^{2} \cdot 14$ .

Factor $4$ out of $56$ .

$x = \frac{- 6 \pm \sqrt{4 (14)}}{2 \cdot 1}$

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

$x = \frac{- 6 \pm \sqrt{2^{2} \cdot 14}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 6 \pm 2 \sqrt{14}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 6 \pm 2 \sqrt{14}}{2}$

Simplify $\frac{- 6 \pm 2 \sqrt{14}}{2}$ .

$x = - 3 \pm \sqrt{14}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - 3 \pm \sqrt{14}$

Decimal Form:

$x = 0.74165738 \dots, - 6.74165738 \dots$