Solve by Completing the Square x^2+10x+22=31

$x^{2} + 10 x + 22 = 31$

Step 1

Move all terms not containing $x$ to the right side of the equation.

Subtract $22$ from both sides of the equation.

$x^{2} + 10 x = 31 - 22$

Subtract $22$ from $31$ .

$x^{2} + 10 x = 9$

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.

$x^{2} + 10 x + {(5)}^{2} = 9 + {(5)}^{2}$

Step 4

Simplify the equation.

Simplify the left side.

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

$x^{2} + 10 x + 25 = 9 + {(5)}^{2}$

Simplify the right side.

Simplify $9 + {(5)}^{2}$ .

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

$x^{2} + 10 x + 25 = 9 + 25$

Add $9$ and $25$ .

$x^{2} + 10 x + 25 = 34$

Step 5

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

${(x + 5)}^{2} = 34$

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.

$x + 5 = \pm \sqrt{34}$

Subtract $5$ from both sides of the equation.

$x = \pm \sqrt{34} - 5$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \pm \sqrt{34} - 5$

Decimal Form:

$x = 0.83095189 \dots, - 10.83095189 \dots$