Pre-Algebra Examples

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

Step 1

Factor each term.

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Step 1.1

Factor by grouping.

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Step 1.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 2 = - 6$ and whose sum is $b = - 5$ .

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Step 1.1.1.1

Factor $- 5$ out of $- 5 m$ .

$\frac{3 m^{2} - 5 (m) - 2}{m^{2} + m - 6} = 2$

Step 1.1.1.2

Rewrite $- 5$ as $1$ plus $- 6$

$\frac{3 m^{2} + (1 - 6) m - 2}{m^{2} + m - 6} = 2$

Step 1.1.1.3

Apply the distributive property.

$\frac{3 m^{2} + 1 m - 6 m - 2}{m^{2} + m - 6} = 2$

Step 1.1.1.4

Multiply $m$ by $1$ .

$\frac{3 m^{2} + m - 6 m - 2}{m^{2} + m - 6} = 2$

Step 1.1.2

Factor out the greatest common factor from each group.

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Step 1.1.2.1

Group the first two terms and the last two terms.

$\frac{(3 m^{2} + m) - 6 m - 2}{m^{2} + m - 6} = 2$

Step 1.1.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{m (3 m + 1) - 2 (3 m + 1)}{m^{2} + m - 6} = 2$

Step 1.1.3

Factor the polynomial by factoring out the greatest common factor, $3 m + 1$ .

$\frac{(3 m + 1) (m - 2)}{m^{2} + m - 6} = 2$

Step 1.2

Factor $m^{2} + m - 6$ using the AC method.

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Step 1.2.1

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 $- 6$ and whose sum is $1$ .

$- 2, 3$

Step 1.2.2

Write the factored form using these integers.

$\frac{(3 m + 1) (m - 2)}{(m - 2) (m + 3)} = 2$

Step 1.3

Reduce the expression $\frac{(3 m + 1) (m - 2)}{(m - 2) (m + 3)}$ by cancelling the common factors.

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Step 1.3.1

Cancel the common factor.

$\frac{(3 m + 1) (m - 2)}{(m - 2) (m + 3)} = 2$

Step 1.3.2

Rewrite the expression.

$\frac{3 m + 1}{m + 3} = 2$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

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

$m + 3, 1$

Step 2.2

Remove parentheses.

$m + 3, 1$

Step 2.3

The LCM of one and any expression is the expression.

$m + 3$

Step 3

Multiply each term in $\frac{3 m + 1}{m + 3} = 2$ by $m + 3$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{3 m + 1}{m + 3} = 2$ by $m + 3$ .

$\frac{3 m + 1}{m + 3} (m + 3) = 2 (m + 3)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $m + 3$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{3 m + 1}{m + 3} (m + 3) = 2 (m + 3)$

Step 3.2.1.2

Rewrite the expression.

$3 m + 1 = 2 (m + 3)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Apply the distributive property.

$3 m + 1 = 2 m + 2 \cdot 3$

Step 3.3.2

Multiply $2$ by $3$ .

$3 m + 1 = 2 m + 6$

Step 4

Solve the equation.

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Step 4.1

Move all terms containing $m$ to the left side of the equation.

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Step 4.1.1

Subtract $2 m$ from both sides of the equation.

$3 m + 1 - 2 m = 6$

Step 4.1.2

Subtract $2 m$ from $3 m$ .

$m + 1 = 6$

Step 4.2

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

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Step 4.2.1

Subtract $1$ from both sides of the equation.

$m = 6 - 1$

Step 4.2.2

Subtract $1$ from $6$ .

$m = 5$