Basic Math Examples

$\frac{2 b}{b + 3} - \frac{3}{b + 1} + \frac{b + 3}{b^{2} + 4 b + 3}$

Step 1

Simplify each term.

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Step 1.1

Factor $b^{2} + 4 b + 3$ 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 $3$ and whose sum is $4$ .

$1, 3$

Step 1.1.2

Write the factored form using these integers.

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

Step 1.2

Cancel the common factor of $b + 3$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

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

Step 2

Combine fractions.

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Step 2.1

Combine the numerators over the common denominator.

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

Step 2.2

Simplify the expression.

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Step 2.2.1

Add $- 3$ and $1$ .

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

Step 2.2.2

Move the negative in front of the fraction.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Factor $2$ out of $2 b (b + 1) - 2 (b + 3)$ .

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Step 7.1.1

Factor $2$ out of $2 b (b + 1)$ .

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

Step 7.1.2

Factor $2$ out of $- 2 (b + 3)$ .

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

Step 7.1.3

Factor $2$ out of $2 (b (b + 1)) + 2 (- (b + 3))$ .

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Multiply $b$ by $b$ .

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

Step 7.4

Multiply $b$ by $1$ .

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

Step 7.5

Apply the distributive property.

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

Step 7.6

Multiply $- 1$ by $3$ .

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

Step 7.7

Subtract $b$ from $b$ .

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

Step 7.8

Add $b^{2}$ and $0$ .

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