Solve by Completing the Square x^2+4x-23=0

$x^{2} + 4 x - 23 = 0$

Step 1

Add $23$ to both sides of the equation.

$x^{2} + 4 x = 23$

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 $b$ .

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

Step 3

Add the term to each side of the equation.

$x^{2} + 4 x + {(2)}^{2} = 23 + {(2)}^{2}$

Step 4

Simplify the equation.

Simplify the left side.

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

$x^{2} + 4 x + 4 = 23 + {(2)}^{2}$

Simplify the right side.

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

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

$x^{2} + 4 x + 4 = 23 + 4$

Add $23$ and $4$ .

$x^{2} + 4 x + 4 = 27$

Step 5

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

${(x + 2)}^{2} = 27$

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.

$x + 2 = \pm \sqrt{27}$

Simplify $\pm \sqrt{27}$ .

Rewrite $27$ as $3^{2} \cdot 3$ .

Factor $9$ out of $27$ .

$x + 2 = \pm \sqrt{9 (3)}$

Rewrite $9$ as $3^{2}$ .

$x + 2 = \pm \sqrt{3^{2} \cdot 3}$

Pull terms out from under the radical.

$x + 2 = \pm 3 \sqrt{3}$

Subtract $2$ from both sides of the equation.

$x = \pm 3 \sqrt{3} - 2$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \pm 3 \sqrt{3} - 2$

Decimal Form:

$x = 3.19615242 \dots, - 7.19615242 \dots$