Solve by Completing the Square x^2+15x=-26

$x^{2} + 15 x = - 26$

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 $b$ .

${(\frac{b}{2})}^{2} = {(\frac{15}{2})}^{2}$

Step 2

Add the term to each side of the equation.

$x^{2} + 15 x + {(\frac{15}{2})}^{2} = - 26 + {(\frac{15}{2})}^{2}$

Step 3

Simplify the equation.

Simplify the left side.

Simplify each term.

Apply the product rule to $\frac{15}{2}$ .

$x^{2} + 15 x + \frac{15^{2}}{2^{2}} = - 26 + {(\frac{15}{2})}^{2}$

Raise $15$ to the power of $2$ .

$x^{2} + 15 x + \frac{225}{2^{2}} = - 26 + {(\frac{15}{2})}^{2}$

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

$x^{2} + 15 x + \frac{225}{4} = - 26 + {(\frac{15}{2})}^{2}$

Simplify the right side.

Simplify $- 26 + {(\frac{15}{2})}^{2}$ .

Simplify each term.

Apply the product rule to $\frac{15}{2}$ .

$x^{2} + 15 x + \frac{225}{4} = - 26 + \frac{15^{2}}{2^{2}}$

Raise $15$ to the power of $2$ .

$x^{2} + 15 x + \frac{225}{4} = - 26 + \frac{225}{2^{2}}$

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

$x^{2} + 15 x + \frac{225}{4} = - 26 + \frac{225}{4}$

To write $- 26$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x^{2} + 15 x + \frac{225}{4} = - 26 \cdot \frac{4}{4} + \frac{225}{4}$

Combine $- 26$ and $\frac{4}{4}$ .

$x^{2} + 15 x + \frac{225}{4} = \frac{- 26 \cdot 4}{4} + \frac{225}{4}$

Combine the numerators over the common denominator.

$x^{2} + 15 x + \frac{225}{4} = \frac{- 26 \cdot 4 + 225}{4}$

Simplify the numerator.

Multiply $- 26$ by $4$ .

$x^{2} + 15 x + \frac{225}{4} = \frac{- 104 + 225}{4}$

Add $- 104$ and $225$ .

$x^{2} + 15 x + \frac{225}{4} = \frac{121}{4}$

Step 4

Factor the perfect trinomial square into ${(x + \frac{15}{2})}^{2}$ .

${(x + \frac{15}{2})}^{2} = \frac{121}{4}$

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.

$x + \frac{15}{2} = \pm \sqrt{\frac{121}{4}}$

Simplify $\pm \sqrt{\frac{121}{4}}$ .

Rewrite $\sqrt{\frac{121}{4}}$ as $\frac{\sqrt{121}}{\sqrt{4}}$ .

$x + \frac{15}{2} = \pm \frac{\sqrt{121}}{\sqrt{4}}$

Simplify the numerator.

Rewrite $121$ as $11^{2}$ .

$x + \frac{15}{2} = \pm \frac{\sqrt{11^{2}}}{\sqrt{4}}$

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

$x + \frac{15}{2} = \pm \frac{11}{\sqrt{4}}$

Simplify the denominator.

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

$x + \frac{15}{2} = \pm \frac{11}{\sqrt{2^{2}}}$

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

$x + \frac{15}{2} = \pm \frac{11}{2}$

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.

$x + \frac{15}{2} = \frac{11}{2}$

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

Subtract $\frac{15}{2}$ from both sides of the equation.

$x = \frac{11}{2} - \frac{15}{2}$

Combine the numerators over the common denominator.

$x = \frac{11 - 15}{2}$

Subtract $15$ from $11$ .

$x = \frac{- 4}{2}$

Divide $- 4$ by $2$ .

$x = - 2$

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

$x + \frac{15}{2} = - \frac{11}{2}$

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

Subtract $\frac{15}{2}$ from both sides of the equation.

$x = - \frac{11}{2} - \frac{15}{2}$

Combine the numerators over the common denominator.

$x = \frac{- 11 - 15}{2}$

Subtract $15$ from $- 11$ .

$x = \frac{- 26}{2}$

Divide $- 26$ by $2$ .

$x = - 13$

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

$x = - 2, - 13$

Solve by Completing the Square x^2+15x=-26

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