Solve by Completing the Square x^2+14x+22=0

$x^{2} + 14 x + 22 = 0$

Step 1

Subtract $22$ from both sides of the equation.

$x^{2} + 14 x = - 22$

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 $b$ .

${(\frac{b}{2})}^{2} = {(7)}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} + 14 x + {(7)}^{2} = - 22 + {(7)}^{2}$

Step 4

Simplify the equation.

Simplify the left side.

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

$x^{2} + 14 x + 49 = - 22 + {(7)}^{2}$

Simplify the right side.

Simplify $- 22 + {(7)}^{2}$ .

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

$x^{2} + 14 x + 49 = - 22 + 49$

Add $- 22$ and $49$ .

$x^{2} + 14 x + 49 = 27$

Step 5

Factor the perfect trinomial square into ${(x + 7)}^{2}$ .

${(x + 7)}^{2} = 27$

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.

$x + 7 = \pm \sqrt{27}$

Simplify $\pm \sqrt{27}$ .

Rewrite $27$ as $3^{2} \cdot 3$ .

Factor $9$ out of $27$ .

$x + 7 = \pm \sqrt{9 (3)}$

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

$x + 7 = \pm \sqrt{3^{2} \cdot 3}$

Pull terms out from under the radical.

$x + 7 = \pm 3 \sqrt{3}$

Subtract $7$ from both sides of the equation.

$x = \pm 3 \sqrt{3} - 7$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \pm 3 \sqrt{3} - 7$

Decimal Form:

$x = - 1.80384757 \dots, - 12.19615242 \dots$