Solve by Completing the Square x^2+14x=-5

$x^{2} + 14 x = - 5$

Step 1

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 2

Add the term to each side of the equation.

$x^{2} + 14 x + {(7)}^{2} = - 5 + {(7)}^{2}$

Step 3

Simplify the equation.

Simplify the left side.

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

$x^{2} + 14 x + 49 = - 5 + {(7)}^{2}$

Simplify the right side.

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

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

$x^{2} + 14 x + 49 = - 5 + 49$

Add $- 5$ and $49$ .

$x^{2} + 14 x + 49 = 44$

Step 4

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

${(x + 7)}^{2} = 44$

Step 5

Solve the equation for $x$ .

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

$x + 7 = \pm \sqrt{44}$

Simplify $\pm \sqrt{44}$ .

Rewrite $44$ as $2^{2} \cdot 11$ .

Factor $4$ out of $44$ .

$x + 7 = \pm \sqrt{4 (11)}$

Rewrite $4$ as $2^{2}$ .

$x + 7 = \pm \sqrt{2^{2} \cdot 11}$

Pull terms out from under the radical.

$x + 7 = \pm 2 \sqrt{11}$

Subtract $7$ from both sides of the equation.

$x = \pm 2 \sqrt{11} - 7$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \pm 2 \sqrt{11} - 7$

Decimal Form:

$x = - 0.36675041 \dots, - 13.63324958 \dots$