Solve by Completing the Square x^2-14x+53=9

$x^{2} - 14 x + 53 = 9$

Step 1

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

Subtract $53$ from both sides of the equation.

$x^{2} - 14 x = 9 - 53$

Subtract $53$ from $9$ .

$x^{2} - 14 x = - 44$

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.

$x^{2} - 14 x + {(- 7)}^{2} = - 44 + {(- 7)}^{2}$

Step 4

Simplify the equation.

Simplify the left side.

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

$x^{2} - 14 x + 49 = - 44 + {(- 7)}^{2}$

Simplify the right side.

Simplify $- 44 + {(- 7)}^{2}$ .

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

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

Add $- 44$ and $49$ .

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

Step 5

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

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

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.

$x - 7 = \pm \sqrt{5}$

Add $7$ to both sides of the equation.

$x = \pm \sqrt{5} + 7$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \pm \sqrt{5} + 7$

Decimal Form:

$x = 9.23606797 \dots, 4.76393202 \dots$