Solve by Completing the Square x^2+3x=0

$x^{2} + 3 x = 0$

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 $b$ .

${(\frac{b}{2})}^{2} = {(\frac{3}{2})}^{2}$

Step 2

Add the term to each side of the equation.

$x^{2} + 3 x + {(\frac{3}{2})}^{2} = 0 + {(\frac{3}{2})}^{2}$

Step 3

Simplify the equation.

Simplify the left side.

Simplify each term.

Apply the product rule to $\frac{3}{2}$ .

$x^{2} + 3 x + \frac{3^{2}}{2^{2}} = 0 + {(\frac{3}{2})}^{2}$

Raise $3$ to the power of $2$ .

$x^{2} + 3 x + \frac{9}{2^{2}} = 0 + {(\frac{3}{2})}^{2}$

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

$x^{2} + 3 x + \frac{9}{4} = 0 + {(\frac{3}{2})}^{2}$

Simplify the right side.

Simplify $0 + {(\frac{3}{2})}^{2}$ .

Simplify each term.

Apply the product rule to $\frac{3}{2}$ .

$x^{2} + 3 x + \frac{9}{4} = 0 + \frac{3^{2}}{2^{2}}$

Raise $3$ to the power of $2$ .

$x^{2} + 3 x + \frac{9}{4} = 0 + \frac{9}{2^{2}}$

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

$x^{2} + 3 x + \frac{9}{4} = 0 + \frac{9}{4}$

Add $0$ and $\frac{9}{4}$ .

$x^{2} + 3 x + \frac{9}{4} = \frac{9}{4}$

Step 4

Factor the perfect trinomial square into ${(x + \frac{3}{2})}^{2}$ .

${(x + \frac{3}{2})}^{2} = \frac{9}{4}$

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.

$x + \frac{3}{2} = \pm \sqrt{\frac{9}{4}}$

Simplify $\pm \sqrt{\frac{9}{4}}$ .

Rewrite $\sqrt{\frac{9}{4}}$ as $\frac{\sqrt{9}}{\sqrt{4}}$ .

$x + \frac{3}{2} = \pm \frac{\sqrt{9}}{\sqrt{4}}$

Simplify the numerator.

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

$x + \frac{3}{2} = \pm \frac{\sqrt{3^{2}}}{\sqrt{4}}$

Pull terms out from under the radical, assuming positive real numbers.

$x + \frac{3}{2} = \pm \frac{3}{\sqrt{4}}$

Simplify the denominator.

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

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

Pull terms out from under the radical, assuming positive real numbers.

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

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x + \frac{3}{2} = \frac{3}{2}$

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

Subtract $\frac{3}{2}$ from both sides of the equation.

$x = \frac{3}{2} - \frac{3}{2}$

Combine the numerators over the common denominator.

$x = \frac{3 - 3}{2}$

Subtract $3$ from $3$ .

$x = \frac{0}{2}$

Divide $0$ by $2$ .

$x = 0$

Next, use the negative value of the $\pm$ to find the second solution.

$x + \frac{3}{2} = - \frac{3}{2}$

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

Subtract $\frac{3}{2}$ from both sides of the equation.

$x = - \frac{3}{2} - \frac{3}{2}$

Combine the numerators over the common denominator.

$x = \frac{- 3 - 3}{2}$

Subtract $3$ from $- 3$ .

$x = \frac{- 6}{2}$

Divide $- 6$ by $2$ .

$x = - 3$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 0, - 3$

Solve by Completing the Square x^2+3x=0

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