Basic Math Examples

$\frac{\frac{y^{2} - 10 y + 25}{y^{2} - 2 y - 35}}{\frac{y^{2} - 25}{9}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{y^{2} - 10 y + 25}{y^{2} - 2 y - 35} \cdot \frac{9}{y^{2} - 25}$

Step 2

Factor using the perfect square rule.

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Step 2.1

Rewrite $25$ as $5^{2}$ .

$\frac{y^{2} - 10 y + 5^{2}}{y^{2} - 2 y - 35} \cdot \frac{9}{y^{2} - 25}$

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.

$10 y = 2 \cdot y \cdot 5$

Step 2.3

Rewrite the polynomial.

$\frac{y^{2} - 2 \cdot y \cdot 5 + 5^{2}}{y^{2} - 2 y - 35} \cdot \frac{9}{y^{2} - 25}$

Step 2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = y$ and $b = 5$ .

$\frac{{(y - 5)}^{2}}{y^{2} - 2 y - 35} \cdot \frac{9}{y^{2} - 25}$

Step 3

Factor $y^{2} - 2 y - 35$ 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 $- 35$ and whose sum is $- 2$ .

$- 7, 5$

Step 3.2

Write the factored form using these integers.

$\frac{{(y - 5)}^{2}}{(y - 7) (y + 5)} \cdot \frac{9}{y^{2} - 25}$

Step 4

Simplify the denominator.

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Step 4.1

Rewrite $25$ as $5^{2}$ .

$\frac{{(y - 5)}^{2}}{(y - 7) (y + 5)} \cdot \frac{9}{y^{2} - 5^{2}}$

Step 4.2

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 $b = 5$ .

$\frac{{(y - 5)}^{2}}{(y - 7) (y + 5)} \cdot \frac{9}{(y + 5) (y - 5)}$

Step 5

Simplify terms.

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Step 5.1

Combine.

$\frac{{(y - 5)}^{2} \cdot 9}{(y - 7) (y + 5) ((y + 5) (y - 5))}$

Step 5.2

Cancel the common factor of ${(y - 5)}^{2}$ and $y - 5$ .

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Step 5.2.1

Factor $y - 5$ out of ${(y - 5)}^{2} \cdot 9$ .

$\frac{(y - 5) ((y - 5) \cdot 9)}{(y - 7) (y + 5) ((y + 5) (y - 5))}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $y - 5$ out of $(y - 7) (y + 5) ((y + 5) (y - 5))$ .

$\frac{(y - 5) ((y - 5) \cdot 9)}{(y - 5) ((y - 7) (y + 5) (y + 5))}$

Step 5.2.2.2

Cancel the common factor.

$\frac{(y - 5) ((y - 5) \cdot 9)}{(y - 5) ((y - 7) (y + 5) (y + 5))}$

Step 5.2.2.3

Rewrite the expression.

$\frac{(y - 5) \cdot 9}{(y - 7) (y + 5) (y + 5)}$

Step 6

Simplify the denominator.

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Step 6.1

Raise $y + 5$ to the power of $1$ .

$\frac{(y - 5) \cdot 9}{(y - 7) ({(y + 5)}^{1} (y + 5))}$

Step 6.2

Raise $y + 5$ to the power of $1$ .

$\frac{(y - 5) \cdot 9}{(y - 7) ({(y + 5)}^{1} {(y + 5)}^{1})}$

Step 6.3

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

$\frac{(y - 5) \cdot 9}{(y - 7) {(y + 5)}^{1 + 1}}$

Step 6.4

Add $1$ and $1$ .

$\frac{(y - 5) \cdot 9}{(y - 7) {(y + 5)}^{2}}$

Step 7

Move $9$ to the left of $y - 5$ .

$\frac{9 (y - 5)}{(y - 7) {(y + 5)}^{2}}$