Basic Math Examples

$\frac{y^{2} - 8 y + 16}{y} \cdot \frac{y + 5}{y^{2} + y - 20}$

Step 1

Factor using the perfect square rule.

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Step 1.1

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

$\frac{y^{2} - 8 y + 4^{2}}{y} \cdot \frac{y + 5}{y^{2} + y - 20}$

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.

$8 y = 2 \cdot y \cdot 4$

Step 1.3

Rewrite the polynomial.

$\frac{y^{2} - 2 \cdot y \cdot 4 + 4^{2}}{y} \cdot \frac{y + 5}{y^{2} + y - 20}$

Step 1.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = y$ and $b = 4$ .

$\frac{{(y - 4)}^{2}}{y} \cdot \frac{y + 5}{y^{2} + y - 20}$

Step 2

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

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

$- 4, 5$

Step 2.2

Write the factored form using these integers.

$\frac{{(y - 4)}^{2}}{y} \cdot \frac{y + 5}{(y - 4) (y + 5)}$

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $y - 4$ .

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Step 3.1.1

Factor $y - 4$ out of ${(y - 4)}^{2}$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Multiply $\frac{y - 4}{y}$ by $\frac{y + 5}{y + 5}$ .

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

Step 3.3

Cancel the common factor of $y + 5$ .

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

$\frac{y - 4}{y}$