Solve by Completing the Square x^2+10x+7=0

$x^{2} + 10 x + 7 = 0$

Step 1

Subtract $7$ from both sides of the equation.

$x^{2} + 10 x = - 7$

Step 2

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(5)}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} + 10 x + {(5)}^{2} = - 7 + {(5)}^{2}$

Step 4

Simplify the equation.

Simplify the left side.

Raise $5$ to the power of $2$ .

$x^{2} + 10 x + 25 = - 7 + {(5)}^{2}$

Simplify the right side.

Simplify $- 7 + {(5)}^{2}$ .

Raise $5$ to the power of $2$ .

$x^{2} + 10 x + 25 = - 7 + 25$

Add $- 7$ and $25$ .

$x^{2} + 10 x + 25 = 18$

Step 5

Factor the perfect trinomial square into ${(x + 5)}^{2}$ .

${(x + 5)}^{2} = 18$

Step 6

Solve the equation for $x$ .

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x + 5 = \pm \sqrt{18}$

Simplify $\pm \sqrt{18}$ .

Rewrite $18$ as $3^{2} \cdot 2$ .

Factor $9$ out of $18$ .

$x + 5 = \pm \sqrt{9 (2)}$

Rewrite $9$ as $3^{2}$ .

$x + 5 = \pm \sqrt{3^{2} \cdot 2}$

Pull terms out from under the radical.

$x + 5 = \pm 3 \sqrt{2}$

Subtract $5$ from both sides of the equation.

$x = \pm 3 \sqrt{2} - 5$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \pm 3 \sqrt{2} - 5$

Decimal Form:

$x = - 0.75735931 \dots, - 9.24264068 \dots$