Finite Math Examples

$4 = \frac{3 x^{2} + 24 x + 36}{5 x + 30}$

Step 1

Rewrite the equation as $\frac{3 x^{2} + 24 x + 36}{5 x + 30} = 4$ .

$\frac{3 x^{2} + 24 x + 36}{5 x + 30} = 4$

Step 2

Factor each term.

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Step 2.1

Factor $3$ out of $3 x^{2} + 24 x + 36$ .

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Step 2.1.1

Factor $3$ out of $3 x^{2}$ .

$\frac{3 (x^{2}) + 24 x + 36}{5 x + 30} = 4$

Step 2.1.2

Factor $3$ out of $24 x$ .

$\frac{3 (x^{2}) + 3 (8 x) + 36}{5 x + 30} = 4$

Step 2.1.3

Factor $3$ out of $36$ .

$\frac{3 x^{2} + 3 (8 x) + 3 \cdot 12}{5 x + 30} = 4$

Step 2.1.4

Factor $3$ out of $3 x^{2} + 3 (8 x)$ .

$\frac{3 (x^{2} + 8 x) + 3 \cdot 12}{5 x + 30} = 4$

Step 2.1.5

Factor $3$ out of $3 (x^{2} + 8 x) + 3 \cdot 12$ .

$\frac{3 (x^{2} + 8 x + 12)}{5 x + 30} = 4$

Step 2.2

Factor.

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Step 2.2.1

Factor $x^{2} + 8 x + 12$ using the AC method.

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Step 2.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $12$ and whose sum is $8$ .

$2, 6$

Step 2.2.1.2

Write the factored form using these integers.

$\frac{3 ((x + 2) (x + 6))}{5 x + 30} = 4$

Step 2.2.2

Remove unnecessary parentheses.

$\frac{3 (x + 2) (x + 6)}{5 x + 30} = 4$

Step 2.3

Factor $5$ out of $5 x + 30$ .

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Step 2.3.1

Factor $5$ out of $5 x$ .

$\frac{3 (x + 2) (x + 6)}{5 (x) + 30} = 4$

Step 2.3.2

Factor $5$ out of $30$ .

$\frac{3 (x + 2) (x + 6)}{5 x + 5 \cdot 6} = 4$

Step 2.3.3

Factor $5$ out of $5 x + 5 \cdot 6$ .

$\frac{3 (x + 2) (x + 6)}{5 (x + 6)} = 4$

Step 2.4

Reduce the expression $\frac{3 (x + 2) (x + 6)}{5 (x + 6)}$ by cancelling the common factors.

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Step 2.4.1

Cancel the common factor.

$\frac{3 (x + 2) (x + 6)}{5 (x + 6)} = 4$

Step 2.4.2

Rewrite the expression.

$\frac{3 (x + 2)}{5} = 4$

Step 3

Find the LCD of the terms in the equation.

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Step 3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$5, 1$

Step 3.2

The LCM of one and any expression is the expression.

$5$

Step 4

Multiply each term in $\frac{3 (x + 2)}{5} = 4$ by $5$ to eliminate the fractions.

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Step 4.1

Multiply each term in $\frac{3 (x + 2)}{5} = 4$ by $5$ .

$\frac{3 (x + 2)}{5} \cdot 5 = 4 \cdot 5$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $5$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{3 (x + 2)}{5} \cdot 5 = 4 \cdot 5$

Step 4.2.1.2

Rewrite the expression.

$3 (x + 2) = 4 \cdot 5$

Step 4.2.2

Apply the distributive property.

$3 x + 3 \cdot 2 = 4 \cdot 5$

Step 4.2.3

Multiply $3$ by $2$ .

$3 x + 6 = 4 \cdot 5$

Step 4.3

Simplify the right side.

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Step 4.3.1

Multiply $4$ by $5$ .

$3 x + 6 = 20$

Step 5

Solve the equation.

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Step 5.1

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

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Step 5.1.1

Subtract $6$ from both sides of the equation.

$3 x = 20 - 6$

Step 5.1.2

Subtract $6$ from $20$ .

$3 x = 14$

Step 5.2

Divide each term in $3 x = 14$ by $3$ and simplify.

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Step 5.2.1

Divide each term in $3 x = 14$ by $3$ .

$\frac{3 x}{3} = \frac{14}{3}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $3$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{14}{3}$

Step 5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{14}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{14}{3}$

Decimal Form:

$x = 4.\overline{6}$

Mixed Number Form:

$x = 4 \frac{2}{3}$