Solve by Completing the Square x^2-10x+23=0

$x^{2} - 10 x + 23 = 0$

Step 1

Subtract $23$ from both sides of the equation.

$x^{2} - 10 x = - 23$

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 $b$ .

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

Step 3

Add the term to each side of the equation.

$x^{2} - 10 x + {(- 5)}^{2} = - 23 + {(- 5)}^{2}$

Step 4

Simplify the equation.

Simplify the left side.

Raise $- 5$ to the power of $2$ .

$x^{2} - 10 x + 25 = - 23 + {(- 5)}^{2}$

Simplify the right side.

Simplify $- 23 + {(- 5)}^{2}$ .

Raise $- 5$ to the power of $2$ .

$x^{2} - 10 x + 25 = - 23 + 25$

Add $- 23$ and $25$ .

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

Step 5

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

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

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.

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

Add $5$ to both sides of the equation.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 6.41421356 \dots, 3.58578643 \dots$