Solve by Completing the Square g^2+7g=144

$g^{2} + 7 g = 144$

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 $b$ .

${(\frac{b}{2})}^{2} = {(\frac{7}{2})}^{2}$

Step 2

Add the term to each side of the equation.

$g^{2} + 7 g + {(\frac{7}{2})}^{2} = 144 + {(\frac{7}{2})}^{2}$

Step 3

Simplify the equation.

Simplify the left side.

Simplify each term.

Apply the product rule to $\frac{7}{2}$ .

$g^{2} + 7 g + \frac{7^{2}}{2^{2}} = 144 + {(\frac{7}{2})}^{2}$

Raise $7$ to the power of $2$ .

$g^{2} + 7 g + \frac{49}{2^{2}} = 144 + {(\frac{7}{2})}^{2}$

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

$g^{2} + 7 g + \frac{49}{4} = 144 + {(\frac{7}{2})}^{2}$

Simplify the right side.

Simplify $144 + {(\frac{7}{2})}^{2}$ .

Simplify each term.

Apply the product rule to $\frac{7}{2}$ .

$g^{2} + 7 g + \frac{49}{4} = 144 + \frac{7^{2}}{2^{2}}$

Raise $7$ to the power of $2$ .

$g^{2} + 7 g + \frac{49}{4} = 144 + \frac{49}{2^{2}}$

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

$g^{2} + 7 g + \frac{49}{4} = 144 + \frac{49}{4}$

To write $144$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$g^{2} + 7 g + \frac{49}{4} = 144 \cdot \frac{4}{4} + \frac{49}{4}$

Combine $144$ and $\frac{4}{4}$ .

$g^{2} + 7 g + \frac{49}{4} = \frac{144 \cdot 4}{4} + \frac{49}{4}$

Combine the numerators over the common denominator.

$g^{2} + 7 g + \frac{49}{4} = \frac{144 \cdot 4 + 49}{4}$

Simplify the numerator.

Multiply $144$ by $4$ .

$g^{2} + 7 g + \frac{49}{4} = \frac{576 + 49}{4}$

Add $576$ and $49$ .

$g^{2} + 7 g + \frac{49}{4} = \frac{625}{4}$

Step 4

Factor the perfect trinomial square into ${(g + \frac{7}{2})}^{2}$ .

${(g + \frac{7}{2})}^{2} = \frac{625}{4}$

Step 5

Solve the equation for $g$ .

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$g + \frac{7}{2} = \pm \sqrt{\frac{625}{4}}$

Simplify $\pm \sqrt{\frac{625}{4}}$ .

Rewrite $\sqrt{\frac{625}{4}}$ as $\frac{\sqrt{625}}{\sqrt{4}}$ .

$g + \frac{7}{2} = \pm \frac{\sqrt{625}}{\sqrt{4}}$

Simplify the numerator.

Rewrite $625$ as $25^{2}$ .

$g + \frac{7}{2} = \pm \frac{\sqrt{25^{2}}}{\sqrt{4}}$

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

$g + \frac{7}{2} = \pm \frac{25}{\sqrt{4}}$

Simplify the denominator.

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

$g + \frac{7}{2} = \pm \frac{25}{\sqrt{2^{2}}}$

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

$g + \frac{7}{2} = \pm \frac{25}{2}$

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.

$g + \frac{7}{2} = \frac{25}{2}$

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

Subtract $\frac{7}{2}$ from both sides of the equation.

$g = \frac{25}{2} - \frac{7}{2}$

Combine the numerators over the common denominator.

$g = \frac{25 - 7}{2}$

Subtract $7$ from $25$ .

$g = \frac{18}{2}$

Divide $18$ by $2$ .

$g = 9$

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

$g + \frac{7}{2} = - \frac{25}{2}$

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

Subtract $\frac{7}{2}$ from both sides of the equation.

$g = - \frac{25}{2} - \frac{7}{2}$

Combine the numerators over the common denominator.

$g = \frac{- 25 - 7}{2}$

Subtract $7$ from $- 25$ .

$g = \frac{- 32}{2}$

Divide $- 32$ by $2$ .

$g = - 16$

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

$g = 9, - 16$

Solve by Completing the Square g^2+7g=144

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