Basic Math Examples

$\frac{2 y}{y^{2} + 5 y + 6} + \frac{y + 1}{y^{2} + 2 y - 3}$

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

Factor $y^{2} + 2 y - 3$ 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.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $y$ by $y$ by adding the exponents.

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Step 6.2.1

Move $y$ .

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

Step 6.2.2

Multiply $y$ by $y$ .

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

Step 6.3

Multiply $- 1$ by $2$ .

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

Step 6.4

Expand $(y + 1) (y + 2)$ using the FOIL Method.

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Step 6.4.1

Apply the distributive property.

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

Step 6.4.2

Apply the distributive property.

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

Step 6.4.3

Apply the distributive property.

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

Step 6.5

Simplify and combine like terms.

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Step 6.5.1

Simplify each term.

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Step 6.5.1.1

Multiply $y$ by $y$ .

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

Step 6.5.1.2

Move $2$ to the left of $y$ .

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

Step 6.5.1.3

Multiply $y$ by $1$ .

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

Step 6.5.1.4

Multiply $2$ by $1$ .

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

Step 6.5.2

Add $2 y$ and $y$ .

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

Step 6.6

Add $2 y^{2}$ and $y^{2}$ .

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

Step 6.7

Add $- 2 y$ and $3 y$ .

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