Solve by Factoring x^2=-7x-7

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

Move all the expressions to the left side of the equation.

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Move $7 x$ to the left side of the equation by adding it to both sides.

$x^{2} + 7 x = - 7$

Move $7$ to the left side of the equation by adding it to both sides.

$x^{2} + 7 x + 7 = 0$

Use the quadratic formula to find the solutions.

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

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

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

Simplify.

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Simplify the numerator.

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

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

Multiply $7$ by $1$ .

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

Multiply $- 4$ by $7$ .

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

Subtract $28$ from $49$ .

$x = \frac{- 7 \pm \sqrt{21}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 7 \pm \sqrt{21}}{2}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Simplify the numerator.

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

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

Multiply $7$ by $1$ .

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

Multiply $- 4$ by $7$ .

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

Subtract $28$ from $49$ .

$x = \frac{- 7 \pm \sqrt{21}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 7 \pm \sqrt{21}}{2}$

Change the $\pm$ to $+$ .

$x = \frac{- 7 + \sqrt{21}}{2}$

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{- 1 \cdot 7 + \sqrt{21}}{2}$

Factor $- 1$ out of $\sqrt{21}$ .

$x = \frac{- 1 \cdot 7 - 1 (- \sqrt{21})}{2}$

Factor $- 1$ out of $- 1 (7) - 1 (- \sqrt{21})$ .

$x = \frac{- 1 (7 - \sqrt{21})}{2}$

Move the negative in front of the fraction.

$x = - \frac{7 - \sqrt{21}}{2}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Simplify the numerator.

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

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

Multiply $7$ by $1$ .

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

Multiply $- 4$ by $7$ .

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

Subtract $28$ from $49$ .

$x = \frac{- 7 \pm \sqrt{21}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 7 \pm \sqrt{21}}{2}$

Change the $\pm$ to $-$ .

$x = \frac{- 7 - \sqrt{21}}{2}$

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{- 1 \cdot 7 - \sqrt{21}}{2}$

Factor $- 1$ out of $- \sqrt{21}$ .

$x = \frac{- 1 \cdot 7 - (\sqrt{21})}{2}$

Factor $- 1$ out of $- 1 (7) - (\sqrt{21})$ .

$x = \frac{- 1 (7 + \sqrt{21})}{2}$

Move the negative in front of the fraction.

$x = - \frac{7 + \sqrt{21}}{2}$

The final answer is the combination of both solutions.

$x = - \frac{7 - \sqrt{21}}{2}, - \frac{7 + \sqrt{21}}{2}$

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{7 - \sqrt{21}}{2}, - \frac{7 + \sqrt{21}}{2}$

Decimal Form:

$x = - 1.20871215 \dots, - 5.79128784 \dots$

Solve by Factoring x^2=-7x-7

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