Solve for x x^2=5-x

$x^{2} = 5 - x$

Add $x$ to both sides of the equation.

$x^{2} + x = 5$

Subtract $5$ from both sides of the equation.

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

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 $c = - 5$ into the quadratic formula and solve for $x$ .

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

Simplify.

Simplify the numerator.

One to any power is one.

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

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $- 5$ .

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

Add $1$ and $20$ .

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

Multiply $2$ by $1$ .

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

The final answer is the combination of both solutions.

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 1.79128784 \dots, - 2.79128784 \dots$