Algebra Examples

$\frac{3 x^{2} - 30 x + 75}{x^{2} - 9 x + 20}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $3$ out of $3 x^{2} - 30 x + 75$ .

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Step 1.1.1

Factor $3$ out of $3 x^{2}$ .

$\frac{3 (x^{2}) - 30 x + 75}{x^{2} - 9 x + 20}$

Step 1.1.2

Factor $3$ out of $- 30 x$ .

$\frac{3 (x^{2}) + 3 (- 10 x) + 75}{x^{2} - 9 x + 20}$

Step 1.1.3

Factor $3$ out of $75$ .

$\frac{3 x^{2} + 3 (- 10 x) + 3 \cdot 25}{x^{2} - 9 x + 20}$

Step 1.1.4

Factor $3$ out of $3 x^{2} + 3 (- 10 x)$ .

$\frac{3 (x^{2} - 10 x) + 3 \cdot 25}{x^{2} - 9 x + 20}$

Step 1.1.5

Factor $3$ out of $3 (x^{2} - 10 x) + 3 \cdot 25$ .

$\frac{3 (x^{2} - 10 x + 25)}{x^{2} - 9 x + 20}$

Step 1.2

Factor using the perfect square rule.

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Step 1.2.1

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

$\frac{3 (x^{2} - 10 x + 5^{2})}{x^{2} - 9 x + 20}$

Step 1.2.2

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

$10 x = 2 \cdot x \cdot 5$

Step 1.2.3

Rewrite the polynomial.

$\frac{3 (x^{2} - 2 \cdot x \cdot 5 + 5^{2})}{x^{2} - 9 x + 20}$

Step 1.2.4

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

$\frac{3 {(x - 5)}^{2}}{x^{2} - 9 x + 20}$

Step 2

Factor $x^{2} - 9 x + 20$ 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 $- 9$ .

$- 5, - 4$

Step 2.2

Write the factored form using these integers.

$\frac{3 {(x - 5)}^{2}}{(x - 5) (x - 4)}$

Step 3

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

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Step 3.1

Factor $x - 5$ out of $3 {(x - 5)}^{2}$ .

$\frac{(x - 5) (3 (x - 5))}{(x - 5) (x - 4)}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Cancel the common factor.

$\frac{(x - 5) (3 (x - 5))}{(x - 5) (x - 4)}$

Step 3.2.2

Rewrite the expression.

$\frac{3 (x - 5)}{x - 4}$