Solve by Completing the Square x^2+5x=3x+11

$x^{2} + 5 x = 3 x + 11$

Step 1

Move all terms containing $x$ to the left side of the equation.

Subtract $3 x$ from both sides of the equation.

$x^{2} + 5 x - 3 x = 11$

Subtract $3 x$ from $5 x$ .

$x^{2} + 2 x = 11$

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 $b$ .

${(\frac{b}{2})}^{2} = {(1)}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} + 2 x + {(1)}^{2} = 11 + {(1)}^{2}$

Step 4

Simplify the equation.

Simplify the left side.

One to any power is one.

$x^{2} + 2 x + 1 = 11 + {(1)}^{2}$

Simplify the right side.

Simplify $11 + {(1)}^{2}$ .

One to any power is one.

$x^{2} + 2 x + 1 = 11 + 1$

Add $11$ and $1$ .

$x^{2} + 2 x + 1 = 12$

Step 5

Factor the perfect trinomial square into ${(x + 1)}^{2}$ .

${(x + 1)}^{2} = 12$

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.

$x + 1 = \pm \sqrt{12}$

Simplify $\pm \sqrt{12}$ .

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$x + 1 = \pm \sqrt{4 (3)}$

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

$x + 1 = \pm \sqrt{2^{2} \cdot 3}$

Pull terms out from under the radical.

$x + 1 = \pm 2 \sqrt{3}$

Subtract $1$ from both sides of the equation.

$x = \pm 2 \sqrt{3} - 1$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \pm 2 \sqrt{3} - 1$

Decimal Form:

$x = 2.46410161 \dots, - 4.46410161 \dots$

Solve by Completing the Square x^2+5x=3x+11

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