Solve for x square root of 6+x-x^2=9

$\sqrt{6 + x - x^{2}} = 9$

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{6 + x - x^{2}}}^{2} = 9^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 + x - x^{2}}$ as ${(6 + x - x^{2})}^{\frac{1}{2}}$ .

${({(6 + x - x^{2})}^{\frac{1}{2}})}^{2} = 9^{2}$

Simplify the left side.

Simplify ${({(6 + x - x^{2})}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(6 + x - x^{2})}^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(6 + x - x^{2})}^{\frac{1}{2} \cdot 2} = 9^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(6 + x - x^{2})}^{\frac{1}{2} \cdot 2} = 9^{2}$

Rewrite the expression.

${(6 + x - x^{2})}^{1} = 9^{2}$

Simplify.

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

Simplify the right side.

Raise $9$ to the power of $2$ .

$6 + x - x^{2} = 81$

Solve for $x$ .

Move all terms to the left side of the equation and simplify.

Subtract $81$ from both sides of the equation.

$6 + x - x^{2} - 81 = 0$

Subtract $81$ from $6$ .

$x - x^{2} - 75 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = - 1$ , $b = 1$ , and $c = - 75$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (- 1 \cdot - 75)}}{2 \cdot - 1}$

Simplify.

Simplify the numerator.

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 1 \cdot - 75}}{2 \cdot - 1}$

Multiply $- 4$ by $- 1$ .

$x = \frac{- 1 \pm \sqrt{1 + 4 \cdot - 75}}{2 \cdot - 1}$

Multiply $4$ by $- 75$ .

$x = \frac{- 1 \pm \sqrt{1 - 300}}{2 \cdot - 1}$

Subtract $300$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 299}}{2 \cdot - 1}$

Rewrite $- 299$ as $- 1 (299)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 299}}{2 \cdot - 1}$

Rewrite $\sqrt{- 1 (299)}$ as $\sqrt{- 1} \cdot \sqrt{299}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{299}}{2 \cdot - 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \sqrt{299}}{2 \cdot - 1}$

Multiply $2$ by $- 1$ .

$x = \frac{- 1 \pm i \sqrt{299}}{- 2}$

Simplify $\frac{- 1 \pm i \sqrt{299}}{- 2}$ .

$x = \frac{1 \pm i \sqrt{299}}{2}$

The final answer is the combination of both solutions.

$x = \frac{1 + i \sqrt{299}}{2}, \frac{1 - i \sqrt{299}}{2}$

Solve for x square root of 6+x-x^2=9

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