Basic Math Examples

$\frac{y^{2} - 14 y + 49}{y^{2} - 49}$

Step 1

Factor using the perfect square rule.

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Step 1.1

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

$\frac{y^{2} - 14 y + 7^{2}}{y^{2} - 49}$

Step 1.2

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 1.3

Rewrite the polynomial.

$\frac{y^{2} - 2 \cdot y \cdot 7 + 7^{2}}{y^{2} - 49}$

Step 1.4

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

$\frac{{(y - 7)}^{2}}{y^{2} - 49}$

Step 2

Simplify the denominator.

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Step 2.1

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

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

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 $b = 7$ .

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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