Solve by Completing the Square x^2-10x=7

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

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 $b$ .

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

Step 2

Add the term to each side of the equation.

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

Step 3

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 $25$ .

$x^{2} - 10 x + 25 = 32$

Step 4

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

${(x - 5)}^{2} = 32$

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.

$x - 5 = \pm \sqrt{32}$

Simplify $\pm \sqrt{32}$ .

Rewrite $32$ as $4^{2} \cdot 2$ .

Factor $16$ out of $32$ .

$x - 5 = \pm \sqrt{16 (2)}$

Rewrite $16$ as $4^{2}$ .

$x - 5 = \pm \sqrt{4^{2} \cdot 2}$

Pull terms out from under the radical.

$x - 5 = \pm 4 \sqrt{2}$

Add $5$ to both sides of the equation.

$x = \pm 4 \sqrt{2} + 5$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \pm 4 \sqrt{2} + 5$

Decimal Form:

$x = 10.65685424 \dots, - 0.65685424 \dots$