Basic Math Examples

$(\frac{y^{2} + 15 y + 56}{y^{2} + 17 y + 72}) - (\frac{y^{2} + 7 y}{y^{2} + 18 y + 81})$

Step 1

Simplify each term.

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Step 1.1

Factor $y^{2} + 15 y + 56$ 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 $56$ and whose sum is $15$ .

$7, 8$

Step 1.1.2

Write the factored form using these integers.

$\frac{(y + 7) (y + 8)}{y^{2} + 17 y + 72} - (\frac{y^{2} + 7 y}{y^{2} + 18 y + 81})$

Step 1.2

Factor $y^{2} + 17 y + 72$ 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 $72$ and whose sum is $17$ .

$8, 9$

Step 1.2.2

Write the factored form using these integers.

$\frac{(y + 7) (y + 8)}{(y + 8) (y + 9)} - (\frac{y^{2} + 7 y}{y^{2} + 18 y + 81})$

Step 1.3

Cancel the common factor of $y + 8$ .

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Step 1.3.1

Cancel the common factor.

$\frac{(y + 7) (y + 8)}{(y + 8) (y + 9)} - (\frac{y^{2} + 7 y}{y^{2} + 18 y + 81})$

Step 1.3.2

Rewrite the expression.

$\frac{y + 7}{y + 9} - (\frac{y^{2} + 7 y}{y^{2} + 18 y + 81})$

Step 1.4

Factor $y$ out of $y^{2} + 7 y$ .

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Step 1.4.1

Factor $y$ out of $y^{2}$ .

$\frac{y + 7}{y + 9} - \frac{y \cdot y + 7 y}{y^{2} + 18 y + 81}$

Step 1.4.2

Factor $y$ out of $7 y$ .

$\frac{y + 7}{y + 9} - \frac{y \cdot y + y \cdot 7}{y^{2} + 18 y + 81}$

Step 1.4.3

Factor $y$ out of $y \cdot y + y \cdot 7$ .

$\frac{y + 7}{y + 9} - \frac{y (y + 7)}{y^{2} + 18 y + 81}$

Step 1.5

Factor using the perfect square rule.

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Step 1.5.1

Rewrite $81$ as $9^{2}$ .

$\frac{y + 7}{y + 9} - \frac{y (y + 7)}{y^{2} + 18 y + 9^{2}}$

Step 1.5.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$18 y = 2 \cdot y \cdot 9$

Step 1.5.3

Rewrite the polynomial.

$\frac{y + 7}{y + 9} - \frac{y (y + 7)}{y^{2} + 2 \cdot y \cdot 9 + 9^{2}}$

Step 1.5.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = y$ and $b = 9$ .

$\frac{y + 7}{y + 9} - \frac{y (y + 7)}{{(y + 9)}^{2}}$

Step 2

To write $\frac{y + 7}{y + 9}$ as a fraction with a common denominator, multiply by $\frac{y + 9}{y + 9}$ .

$\frac{y + 7}{y + 9} \cdot \frac{y + 9}{y + 9} - \frac{y (y + 7)}{{(y + 9)}^{2}}$

Step 3

Write each expression with a common denominator of ${(y + 9)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{y + 7}{y + 9}$ by $\frac{y + 9}{y + 9}$ .

$\frac{(y + 7) (y + 9)}{(y + 9) (y + 9)} - \frac{y (y + 7)}{{(y + 9)}^{2}}$

Step 3.2

Raise $y + 9$ to the power of $1$ .

$\frac{(y + 7) (y + 9)}{{(y + 9)}^{1} (y + 9)} - \frac{y (y + 7)}{{(y + 9)}^{2}}$

Step 3.3

Raise $y + 9$ to the power of $1$ .

$\frac{(y + 7) (y + 9)}{{(y + 9)}^{1} {(y + 9)}^{1}} - \frac{y (y + 7)}{{(y + 9)}^{2}}$

Step 3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(y + 7) (y + 9)}{{(y + 9)}^{1 + 1}} - \frac{y (y + 7)}{{(y + 9)}^{2}}$

Step 3.5

Add $1$ and $1$ .

$\frac{(y + 7) (y + 9)}{{(y + 9)}^{2}} - \frac{y (y + 7)}{{(y + 9)}^{2}}$

Step 4

Combine the numerators over the common denominator.

$\frac{(y + 7) (y + 9) - y (y + 7)}{{(y + 9)}^{2}}$

Step 5

Simplify the numerator.

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Step 5.1

Factor $y + 7$ out of $(y + 7) (y + 9) - y (y + 7)$ .

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Step 5.1.1

Factor $y + 7$ out of $- y (y + 7)$ .

$\frac{(y + 7) (y + 9) + (y + 7) (- y)}{{(y + 9)}^{2}}$

Step 5.1.2

Factor $y + 7$ out of $(y + 7) (y + 9) + (y + 7) (- y)$ .

$\frac{(y + 7) (y + 9 - y)}{{(y + 9)}^{2}}$

Step 5.2

Subtract $y$ from $y$ .

$\frac{(y + 7) (0 + 9)}{{(y + 9)}^{2}}$

Step 5.3

Add $0$ and $9$ .

$\frac{(y + 7) \cdot 9}{{(y + 9)}^{2}}$

Step 6

Move $9$ to the left of $y + 7$ .

$\frac{9 (y + 7)}{{(y + 9)}^{2}}$