Finite Math Examples

$9 x^{2} + 15 x = 14$

Step 1

Divide each term in $9 x^{2} + 15 x = 14$ by $9$ and simplify.

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Step 1.1

Divide each term in $9 x^{2} + 15 x = 14$ by $9$ .

$\frac{9 x^{2}}{9} + \frac{15 x}{9} = \frac{14}{9}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Cancel the common factor of $9$ .

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Step 1.2.1.1.1

Cancel the common factor.

$\frac{9 x^{2}}{9} + \frac{15 x}{9} = \frac{14}{9}$

Step 1.2.1.1.2

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

$x^{2} + \frac{15 x}{9} = \frac{14}{9}$

Step 1.2.1.2

Cancel the common factor of $15$ and $9$ .

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Step 1.2.1.2.1

Factor $3$ out of $15 x$ .

$x^{2} + \frac{3 (5 x)}{9} = \frac{14}{9}$

Step 1.2.1.2.2

Cancel the common factors.

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Step 1.2.1.2.2.1

Factor $3$ out of $9$ .

$x^{2} + \frac{3 (5 x)}{3 (3)} = \frac{14}{9}$

Step 1.2.1.2.2.2

Cancel the common factor.

$x^{2} + \frac{3 (5 x)}{3 \cdot 3} = \frac{14}{9}$

Step 1.2.1.2.2.3

Rewrite the expression.

$x^{2} + \frac{5 x}{3} = \frac{14}{9}$

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 $b$ .

${(\frac{b}{2})}^{2} = {(\frac{5}{6})}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} + \frac{5 x}{3} + {(\frac{5}{6})}^{2} = \frac{14}{9} + {(\frac{5}{6})}^{2}$

Step 4

Simplify the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify each term.

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Step 4.1.1.1

Apply the product rule to $\frac{5}{6}$ .

$x^{2} + \frac{5 x}{3} + \frac{5^{2}}{6^{2}} = \frac{14}{9} + {(\frac{5}{6})}^{2}$

Step 4.1.1.2

Raise $5$ to the power of $2$ .

$x^{2} + \frac{5 x}{3} + \frac{25}{6^{2}} = \frac{14}{9} + {(\frac{5}{6})}^{2}$

Step 4.1.1.3

Raise $6$ to the power of $2$ .

$x^{2} + \frac{5 x}{3} + \frac{25}{36} = \frac{14}{9} + {(\frac{5}{6})}^{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{14}{9} + {(\frac{5}{6})}^{2}$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Apply the product rule to $\frac{5}{6}$ .

$x^{2} + \frac{5 x}{3} + \frac{25}{36} = \frac{14}{9} + \frac{5^{2}}{6^{2}}$

Step 4.2.1.1.2

Raise $5$ to the power of $2$ .

$x^{2} + \frac{5 x}{3} + \frac{25}{36} = \frac{14}{9} + \frac{25}{6^{2}}$

Step 4.2.1.1.3

Raise $6$ to the power of $2$ .

$x^{2} + \frac{5 x}{3} + \frac{25}{36} = \frac{14}{9} + \frac{25}{36}$

Step 4.2.1.2

To write $\frac{14}{9}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x^{2} + \frac{5 x}{3} + \frac{25}{36} = \frac{14}{9} \cdot \frac{4}{4} + \frac{25}{36}$

Step 4.2.1.3

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.1.3.1

Multiply $\frac{14}{9}$ by $\frac{4}{4}$ .

$x^{2} + \frac{5 x}{3} + \frac{25}{36} = \frac{14 \cdot 4}{9 \cdot 4} + \frac{25}{36}$

Step 4.2.1.3.2

Multiply $9$ by $4$ .

$x^{2} + \frac{5 x}{3} + \frac{25}{36} = \frac{14 \cdot 4}{36} + \frac{25}{36}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$x^{2} + \frac{5 x}{3} + \frac{25}{36} = \frac{14 \cdot 4 + 25}{36}$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $14$ by $4$ .

$x^{2} + \frac{5 x}{3} + \frac{25}{36} = \frac{56 + 25}{36}$

Step 4.2.1.5.2

Add $56$ and $25$ .

$x^{2} + \frac{5 x}{3} + \frac{25}{36} = \frac{81}{36}$

Step 4.2.1.6

Cancel the common factor of $81$ and $36$ .

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Step 4.2.1.6.1

Factor $9$ out of $81$ .

$x^{2} + \frac{5 x}{3} + \frac{25}{36} = \frac{9 (9)}{36}$

Step 4.2.1.6.2

Cancel the common factors.

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Step 4.2.1.6.2.1

Factor $9$ out of $36$ .

$x^{2} + \frac{5 x}{3} + \frac{25}{36} = \frac{9 \cdot 9}{9 \cdot 4}$

Step 4.2.1.6.2.2

Cancel the common factor.

$x^{2} + \frac{5 x}{3} + \frac{25}{36} = \frac{9 \cdot 9}{9 \cdot 4}$

Step 4.2.1.6.2.3

Rewrite the expression.

$x^{2} + \frac{5 x}{3} + \frac{25}{36} = \frac{9}{4}$

Step 5

Factor the perfect trinomial square into ${(x + \frac{5}{6})}^{2}$ .

${(x + \frac{5}{6})}^{2} = \frac{9}{4}$

Step 6

Solve the equation for $x$ .

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Step 6.1

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

$x + \frac{5}{6} = \pm \sqrt{\frac{9}{4}}$

Step 6.2

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

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Step 6.2.1

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

$x + \frac{5}{6} = \pm \frac{\sqrt{9}}{\sqrt{4}}$

Step 6.2.2

Simplify the numerator.

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Step 6.2.2.1

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

$x + \frac{5}{6} = \pm \frac{\sqrt{3^{2}}}{\sqrt{4}}$

Step 6.2.2.2

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

$x + \frac{5}{6} = \pm \frac{3}{\sqrt{4}}$

Step 6.2.3

Simplify the denominator.

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Step 6.2.3.1

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

$x + \frac{5}{6} = \pm \frac{3}{\sqrt{2^{2}}}$

Step 6.2.3.2

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

$x + \frac{5}{6} = \pm \frac{3}{2}$

Step 6.3

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

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Step 6.3.1

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

$x + \frac{5}{6} = \frac{3}{2}$

Step 6.3.2

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

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Step 6.3.2.1

Subtract $\frac{5}{6}$ from both sides of the equation.

$x = \frac{3}{2} - \frac{5}{6}$

Step 6.3.2.2

To write $\frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = \frac{3}{2} \cdot \frac{3}{3} - \frac{5}{6}$

Step 6.3.2.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.3.2.3.1

Multiply $\frac{3}{2}$ by $\frac{3}{3}$ .

$x = \frac{3 \cdot 3}{2 \cdot 3} - \frac{5}{6}$

Step 6.3.2.3.2

Multiply $2$ by $3$ .

$x = \frac{3 \cdot 3}{6} - \frac{5}{6}$

Step 6.3.2.4

Combine the numerators over the common denominator.

$x = \frac{3 \cdot 3 - 5}{6}$

Step 6.3.2.5

Simplify the numerator.

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Step 6.3.2.5.1

Multiply $3$ by $3$ .

$x = \frac{9 - 5}{6}$

Step 6.3.2.5.2

Subtract $5$ from $9$ .

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

Step 6.3.2.6

Cancel the common factor of $4$ and $6$ .

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Step 6.3.2.6.1

Factor $2$ out of $4$ .

$x = \frac{2 (2)}{6}$

Step 6.3.2.6.2

Cancel the common factors.

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Step 6.3.2.6.2.1

Factor $2$ out of $6$ .

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

Step 6.3.2.6.2.2

Cancel the common factor.

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

Step 6.3.2.6.2.3

Rewrite the expression.

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

Step 6.3.3

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

$x + \frac{5}{6} = - \frac{3}{2}$

Step 6.3.4

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

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Step 6.3.4.1

Subtract $\frac{5}{6}$ from both sides of the equation.

$x = - \frac{3}{2} - \frac{5}{6}$

Step 6.3.4.2

To write $- \frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x = - \frac{3}{2} \cdot \frac{3}{3} - \frac{5}{6}$

Step 6.3.4.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.3.4.3.1

Multiply $\frac{3}{2}$ by $\frac{3}{3}$ .

$x = - \frac{3 \cdot 3}{2 \cdot 3} - \frac{5}{6}$

Step 6.3.4.3.2

Multiply $2$ by $3$ .

$x = - \frac{3 \cdot 3}{6} - \frac{5}{6}$

Step 6.3.4.4

Combine the numerators over the common denominator.

$x = \frac{- 3 \cdot 3 - 5}{6}$

Step 6.3.4.5

Simplify the numerator.

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Step 6.3.4.5.1

Multiply $- 3$ by $3$ .

$x = \frac{- 9 - 5}{6}$

Step 6.3.4.5.2

Subtract $5$ from $- 9$ .

$x = \frac{- 14}{6}$

Step 6.3.4.6

Cancel the common factor of $- 14$ and $6$ .

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Step 6.3.4.6.1

Factor $2$ out of $- 14$ .

$x = \frac{2 (- 7)}{6}$

Step 6.3.4.6.2

Cancel the common factors.

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Step 6.3.4.6.2.1

Factor $2$ out of $6$ .

$x = \frac{2 \cdot - 7}{2 \cdot 3}$

Step 6.3.4.6.2.2

Cancel the common factor.

$x = \frac{2 \cdot - 7}{2 \cdot 3}$

Step 6.3.4.6.2.3

Rewrite the expression.

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

Step 6.3.4.7

Move the negative in front of the fraction.

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

Step 6.3.5

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

$x = \frac{2}{3}, - \frac{7}{3}$