Pre-Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Factor $x^{2} + 5 x + 6$ using the AC method.

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Step 1.1.1

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 $6$ and whose sum is $5$ .

$2, 3$

Step 1.1.2

Write the factored form using these integers.

$\frac{3 x}{(x + 2) (x + 3)} + \frac{x - 2}{x^{2} + 2 x - 3}$

Step 1.2

Factor $x^{2} + 2 x - 3$ 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 $- 3$ and whose sum is $2$ .

$- 1, 3$

Step 1.2.2

Write the factored form using these integers.

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

Step 2

To write $\frac{3 x}{(x + 2) (x + 3)}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{3 x}{(x + 2) (x + 3)} \cdot \frac{x - 1}{x - 1} + \frac{x - 2}{(x - 1) (x + 3)}$

Step 3

To write $\frac{x - 2}{(x - 1) (x + 3)}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$\frac{3 x}{(x + 2) (x + 3)} \cdot \frac{x - 1}{x - 1} + \frac{x - 2}{(x - 1) (x + 3)} \cdot \frac{x + 2}{x + 2}$

Step 4

Write each expression with a common denominator of $(x + 2) (x + 3) (x - 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{3 x}{(x + 2) (x + 3)}$ by $\frac{x - 1}{x - 1}$ .

$\frac{3 x (x - 1)}{(x + 2) (x + 3) (x - 1)} + \frac{x - 2}{(x - 1) (x + 3)} \cdot \frac{x + 2}{x + 2}$

Step 4.2

Multiply $\frac{x - 2}{(x - 1) (x + 3)}$ by $\frac{x + 2}{x + 2}$ .

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

Step 4.3

Reorder the factors of $(x - 1) (x + 3) (x + 2)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{3 x \cdot x + 3 x \cdot - 1 + (x - 2) (x + 2)}{(x + 2) (x + 3) (x - 1)}$

Step 6.2

Multiply $x$ by $x$ by adding the exponents.

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Step 6.2.1

Move $x$ .

$\frac{3 (x \cdot x) + 3 x \cdot - 1 + (x - 2) (x + 2)}{(x + 2) (x + 3) (x - 1)}$

Step 6.2.2

Multiply $x$ by $x$ .

$\frac{3 x^{2} + 3 x \cdot - 1 + (x - 2) (x + 2)}{(x + 2) (x + 3) (x - 1)}$

Step 6.3

Multiply $- 1$ by $3$ .

$\frac{3 x^{2} - 3 x + (x - 2) (x + 2)}{(x + 2) (x + 3) (x - 1)}$

Step 6.4

Expand $(x - 2) (x + 2)$ using the FOIL Method.

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Step 6.4.1

Apply the distributive property.

$\frac{3 x^{2} - 3 x + x (x + 2) - 2 (x + 2)}{(x + 2) (x + 3) (x - 1)}$

Step 6.4.2

Apply the distributive property.

$\frac{3 x^{2} - 3 x + x \cdot x + x \cdot 2 - 2 (x + 2)}{(x + 2) (x + 3) (x - 1)}$

Step 6.4.3

Apply the distributive property.

$\frac{3 x^{2} - 3 x + x \cdot x + x \cdot 2 - 2 x - 2 \cdot 2}{(x + 2) (x + 3) (x - 1)}$

Step 6.5

Combine the opposite terms in $x \cdot x + x \cdot 2 - 2 x - 2 \cdot 2$ .

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Step 6.5.1

Reorder the factors in the terms $x \cdot 2$ and $- 2 x$ .

$\frac{3 x^{2} - 3 x + x \cdot x + 2 x - 2 x - 2 \cdot 2}{(x + 2) (x + 3) (x - 1)}$

Step 6.5.2

Subtract $2 x$ from $2 x$ .

$\frac{3 x^{2} - 3 x + x \cdot x + 0 - 2 \cdot 2}{(x + 2) (x + 3) (x - 1)}$

Step 6.5.3

Add $x \cdot x$ and $0$ .

$\frac{3 x^{2} - 3 x + x \cdot x - 2 \cdot 2}{(x + 2) (x + 3) (x - 1)}$

Step 6.6

Simplify each term.

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Step 6.6.1

Multiply $x$ by $x$ .

$\frac{3 x^{2} - 3 x + x^{2} - 2 \cdot 2}{(x + 2) (x + 3) (x - 1)}$

Step 6.6.2

Multiply $- 2$ by $2$ .

$\frac{3 x^{2} - 3 x + x^{2} - 4}{(x + 2) (x + 3) (x - 1)}$

Step 6.7

Add $3 x^{2}$ and $x^{2}$ .

$\frac{4 x^{2} - 3 x - 4}{(x + 2) (x + 3) (x - 1)}$