Basic Math Examples

$\frac{3 y^{2} - 6}{y^{2} + y - 20} + \frac{y - 9}{y^{2} + y - 20} - \frac{2 y^{2} + y + 1}{y^{2} + y - 20}$

Step 1

Simplify terms.

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Step 1.1

Combine the numerators over the common denominator.

$\frac{3 y^{2} - 6 + y - 9 - (2 y^{2} + y + 1)}{y^{2} + y - 20}$

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the distributive property.

$\frac{3 y^{2} - 6 + y - 9 - (2 y^{2}) - y - 1 \cdot 1}{y^{2} + y - 20}$

Step 1.2.2

Simplify.

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Step 1.2.2.1

Multiply $2$ by $- 1$ .

$\frac{3 y^{2} - 6 + y - 9 - 2 y^{2} - y - 1 \cdot 1}{y^{2} + y - 20}$

Step 1.2.2.2

Multiply $- 1$ by $1$ .

$\frac{3 y^{2} - 6 + y - 9 - 2 y^{2} - y - 1}{y^{2} + y - 20}$

Step 1.3

Simplify by adding terms.

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Step 1.3.1

Combine the opposite terms in $3 y^{2} - 6 + y - 9 - 2 y^{2} - y - 1$ .

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Step 1.3.1.1

Subtract $y$ from $y$ .

$\frac{3 y^{2} - 6 - 9 - 2 y^{2} + 0 - 1}{y^{2} + y - 20}$

Step 1.3.1.2

Add $3 y^{2} - 6 - 9 - 2 y^{2}$ and $0$ .

$\frac{3 y^{2} - 6 - 9 - 2 y^{2} - 1}{y^{2} + y - 20}$

Step 1.3.2

Subtract $2 y^{2}$ from $3 y^{2}$ .

$\frac{y^{2} - 6 - 9 - 1}{y^{2} + y - 20}$

Step 1.3.3

Simplify by subtracting numbers.

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Step 1.3.3.1

Subtract $9$ from $- 6$ .

$\frac{y^{2} - 15 - 1}{y^{2} + y - 20}$

Step 1.3.3.2

Subtract $1$ from $- 15$ .

$\frac{y^{2} - 16}{y^{2} + y - 20}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $16$ as $4^{2}$ .

$\frac{y^{2} - 4^{2}}{y^{2} + y - 20}$

Step 2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = 4$ .

$\frac{(y + 4) (y - 4)}{y^{2} + y - 20}$

Step 3

Factor $y^{2} + y - 20$ using the AC method.

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Step 3.1

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 $- 20$ and whose sum is $1$ .

$- 4, 5$

Step 3.2

Write the factored form using these integers.

$\frac{(y + 4) (y - 4)}{(y - 4) (y + 5)}$

Step 4

Cancel the common factor of $y - 4$ .

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Step 4.1

Cancel the common factor.

$\frac{(y + 4) (y - 4)}{(y - 4) (y + 5)}$

Step 4.2

Rewrite the expression.

$\frac{y + 4}{y + 5}$