Basic Math Examples

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

Step 1

Simplify each term.

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Step 1.1

Factor $b^{2} + 5 b + 6$ 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.

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

Step 1.2

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

$1, 2$

Step 1.2.2

Write the factored form using these integers.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $b$ by $b$ .

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

Step 6.3

Multiply $b$ by $1$ .

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

Step 6.4

Apply the distributive property.

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

Step 6.5

Multiply $- 2$ by $3$ .

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

Step 6.6

Subtract $2 b$ from $b$ .

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

Step 6.7

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

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Step 6.7.1

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 $- 1$ .

$- 3, 2$

Step 6.7.2

Write the factored form using these integers.

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

Step 7

Cancel the common factor of $b + 2$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

$\frac{b - 3}{(b + 1) (b + 3)}$