Solve by Completing the Square x^2-14x=7

$x^{2} - 14 x = 7$

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.

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

Step 3

Simplify the equation.

Simplify the left side.

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

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

Simplify the right side.

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

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

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

Add $7$ and $49$ .

$x^{2} - 14 x + 49 = 56$

Step 4

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

${(x - 7)}^{2} = 56$

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.

$x - 7 = \pm \sqrt{56}$

Simplify $\pm \sqrt{56}$ .

Rewrite $56$ as $2^{2} \cdot 14$ .

Factor $4$ out of $56$ .

$x - 7 = \pm \sqrt{4 (14)}$

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

$x - 7 = \pm \sqrt{2^{2} \cdot 14}$

Pull terms out from under the radical.

$x - 7 = \pm 2 \sqrt{14}$

Add $7$ to both sides of the equation.

$x = \pm 2 \sqrt{14} + 7$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \pm 2 \sqrt{14} + 7$

Decimal Form:

$x = 14.48331477 \dots, - 0.48331477 \dots$