Basic Math Examples

$\frac{y^{2} + 11 y + 28}{y^{2} + 14 y + 49}$

Step 1

Factor $y^{2} + 11 y + 28$ using the AC method.

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Step 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 $28$ and whose sum is $11$ .

$4, 7$

Step 1.2

Write the factored form using these integers.

$\frac{(y + 4) (y + 7)}{y^{2} + 14 y + 49}$

Step 2

Factor using the perfect square rule.

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Step 2.1

Rewrite $49$ as $7^{2}$ .

$\frac{(y + 4) (y + 7)}{y^{2} + 14 y + 7^{2}}$

Step 2.2

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

$14 y = 2 \cdot y \cdot 7$

Step 2.3

Rewrite the polynomial.

$\frac{(y + 4) (y + 7)}{y^{2} + 2 \cdot y \cdot 7 + 7^{2}}$

Step 2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = y$ and $b = 7$ .

$\frac{(y + 4) (y + 7)}{{(y + 7)}^{2}}$

Step 3

Cancel the common factor of $y + 7$ and ${(y + 7)}^{2}$ .

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Step 3.1

Factor $y + 7$ out of $(y + 4) (y + 7)$ .

$\frac{(y + 7) (y + 4)}{{(y + 7)}^{2}}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

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

$\frac{(y + 7) (y + 4)}{(y + 7) (y + 7)}$

Step 3.2.2

Cancel the common factor.

$\frac{(y + 7) (y + 4)}{(y + 7) (y + 7)}$

Step 3.2.3

Rewrite the expression.

$\frac{y + 4}{y + 7}$