Solve for x 6x+9=x^2

$6 x + 9 = x^{2}$

Subtract $x^{2}$ from both sides of the equation.

$6 x + 9 - x^{2} = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = - 1$ , $b = 6$ , and $c = 9$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (- 1 \cdot 9)}}{2 \cdot - 1}$

Simplify.

Simplify the numerator.

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

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 1 \cdot 9}}{2 \cdot - 1}$

Multiply $- 4$ by $- 1$ .

$x = \frac{- 6 \pm \sqrt{36 + 4 \cdot 9}}{2 \cdot - 1}$

Multiply $4$ by $9$ .

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

Add $36$ and $36$ .

$x = \frac{- 6 \pm \sqrt{72}}{2 \cdot - 1}$

Rewrite $72$ as $6^{2} \cdot 2$ .

Factor $36$ out of $72$ .

$x = \frac{- 6 \pm \sqrt{36 (2)}}{2 \cdot - 1}$

Rewrite $36$ as $6^{2}$ .

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 1$ .

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

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

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

The final answer is the combination of both solutions.

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 7.24264068 \dots, - 1.24264068 \dots$