Solve by Completing the Square 12x^2+25x=7

$12 x^{2} + 25 x = 7$

Step 1

Divide each term in $12 x^{2} + 25 x = 7$ by $12$ and simplify.

Divide each term in $12 x^{2} + 25 x = 7$ by $12$ .

$\frac{12 x^{2}}{12} + \frac{25 x}{12} = \frac{7}{12}$

Simplify the left side.

Cancel the common factor of $12$ .

Cancel the common factor.

$\frac{12 x^{2}}{12} + \frac{25 x}{12} = \frac{7}{12}$

Divide $x^{2}$ by $1$ .

$x^{2} + \frac{25 x}{12} = \frac{7}{12}$

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 $b$ .

${(\frac{b}{2})}^{2} = {(\frac{25}{24})}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} + \frac{25 x}{12} + {(\frac{25}{24})}^{2} = \frac{7}{12} + {(\frac{25}{24})}^{2}$

Step 4

Simplify the equation.

Simplify the left side.

Simplify each term.

Apply the product rule to $\frac{25}{24}$ .

$x^{2} + \frac{25 x}{12} + \frac{25^{2}}{24^{2}} = \frac{7}{12} + {(\frac{25}{24})}^{2}$

Raise $25$ to the power of $2$ .

$x^{2} + \frac{25 x}{12} + \frac{625}{24^{2}} = \frac{7}{12} + {(\frac{25}{24})}^{2}$

Raise $24$ to the power of $2$ .

$x^{2} + \frac{25 x}{12} + \frac{625}{576} = \frac{7}{12} + {(\frac{25}{24})}^{2}$

Simplify the right side.

Simplify $\frac{7}{12} + {(\frac{25}{24})}^{2}$ .

Simplify each term.

Apply the product rule to $\frac{25}{24}$ .

$x^{2} + \frac{25 x}{12} + \frac{625}{576} = \frac{7}{12} + \frac{25^{2}}{24^{2}}$

Raise $25$ to the power of $2$ .

$x^{2} + \frac{25 x}{12} + \frac{625}{576} = \frac{7}{12} + \frac{625}{24^{2}}$

Raise $24$ to the power of $2$ .

$x^{2} + \frac{25 x}{12} + \frac{625}{576} = \frac{7}{12} + \frac{625}{576}$

To write $\frac{7}{12}$ as a fraction with a common denominator, multiply by $\frac{48}{48}$ .

$x^{2} + \frac{25 x}{12} + \frac{625}{576} = \frac{7}{12} \cdot \frac{48}{48} + \frac{625}{576}$

Write each expression with a common denominator of $576$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{7}{12}$ by $\frac{48}{48}$ .

$x^{2} + \frac{25 x}{12} + \frac{625}{576} = \frac{7 \cdot 48}{12 \cdot 48} + \frac{625}{576}$

Multiply $12$ by $48$ .

$x^{2} + \frac{25 x}{12} + \frac{625}{576} = \frac{7 \cdot 48}{576} + \frac{625}{576}$

Combine the numerators over the common denominator.

$x^{2} + \frac{25 x}{12} + \frac{625}{576} = \frac{7 \cdot 48 + 625}{576}$

Simplify the numerator.

Multiply $7$ by $48$ .

$x^{2} + \frac{25 x}{12} + \frac{625}{576} = \frac{336 + 625}{576}$

Add $336$ and $625$ .

$x^{2} + \frac{25 x}{12} + \frac{625}{576} = \frac{961}{576}$

Step 5

Factor the perfect trinomial square into ${(x + \frac{25}{24})}^{2}$ .

${(x + \frac{25}{24})}^{2} = \frac{961}{576}$

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.

$x + \frac{25}{24} = \pm \sqrt{\frac{961}{576}}$

Simplify $\pm \sqrt{\frac{961}{576}}$ .

Rewrite $\sqrt{\frac{961}{576}}$ as $\frac{\sqrt{961}}{\sqrt{576}}$ .

$x + \frac{25}{24} = \pm \frac{\sqrt{961}}{\sqrt{576}}$

Simplify the numerator.

Rewrite $961$ as $31^{2}$ .

$x + \frac{25}{24} = \pm \frac{\sqrt{31^{2}}}{\sqrt{576}}$

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

$x + \frac{25}{24} = \pm \frac{31}{\sqrt{576}}$

Simplify the denominator.

Rewrite $576$ as $24^{2}$ .

$x + \frac{25}{24} = \pm \frac{31}{\sqrt{24^{2}}}$

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

$x + \frac{25}{24} = \pm \frac{31}{24}$

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.

$x + \frac{25}{24} = \frac{31}{24}$

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

Subtract $\frac{25}{24}$ from both sides of the equation.

$x = \frac{31}{24} - \frac{25}{24}$

Combine the numerators over the common denominator.

$x = \frac{31 - 25}{24}$

Subtract $25$ from $31$ .

$x = \frac{6}{24}$

Cancel the common factor of $6$ and $24$ .

Factor $6$ out of $6$ .

$x = \frac{6 (1)}{24}$

Cancel the common factors.

Factor $6$ out of $24$ .

$x = \frac{6 \cdot 1}{6 \cdot 4}$

Cancel the common factor.

$x = \frac{6 \cdot 1}{6 \cdot 4}$

Rewrite the expression.

$x = \frac{1}{4}$

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

$x + \frac{25}{24} = - \frac{31}{24}$

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

Subtract $\frac{25}{24}$ from both sides of the equation.

$x = - \frac{31}{24} - \frac{25}{24}$

Combine the numerators over the common denominator.

$x = \frac{- 31 - 25}{24}$

Subtract $25$ from $- 31$ .

$x = \frac{- 56}{24}$

Cancel the common factor of $- 56$ and $24$ .

Factor $8$ out of $- 56$ .

$x = \frac{8 (- 7)}{24}$

Cancel the common factors.

Factor $8$ out of $24$ .

$x = \frac{8 \cdot - 7}{8 \cdot 3}$

Cancel the common factor.

$x = \frac{8 \cdot - 7}{8 \cdot 3}$

Rewrite the expression.

$x = \frac{- 7}{3}$

Move the negative in front of the fraction.

$x = - \frac{7}{3}$

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

$x = \frac{1}{4}, - \frac{7}{3}$

Solve by Completing the Square 12x^2+25x=7

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