Basic Math Examples

\frac{x^{2} - 7 x + 6}{5 x^{2} - 5}

$\frac{x^{2} - 7 x + 6}{5 x^{2} - 5}$

Step 1

Factor

x^{2} - 7 x + 6

$x^{2} - 7 x + 6$ using the AC method.

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Step 1.1

Consider the form

x^{2} + b x + c

$x^{2} + b x + c$ . Find a pair of integers whose product is

c

$c$ and whose sum is

b

$b$ . In this case, whose product is

6

$6$ and whose sum is

- 7

$- 7$ .

- 6, - 1

$- 6, - 1$

Step 1.2

Write the factored form using these integers.

\frac{(x - 6) (x - 1)}{5 x^{2} - 5}

$\frac{(x - 6) (x - 1)}{5 x^{2} - 5}$

\frac{(x - 6) (x - 1)}{5 x^{2} - 5}

$\frac{(x - 6) (x - 1)}{5 x^{2} - 5}$

Step 2

Simplify the denominator.

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Step 2.1

Factor

5

$5$ out of

5 x^{2} - 5

$5 x^{2} - 5$ .

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Step 2.1.1

Factor

5

$5$ out of

5 x^{2}

$5 x^{2}$ .

\frac{(x - 6) (x - 1)}{5 (x^{2}) - 5}

$\frac{(x - 6) (x - 1)}{5 (x^{2}) - 5}$

Step 2.1.2

Factor

5

$5$ out of

- 5

$- 5$ .

\frac{(x - 6) (x - 1)}{5 (x^{2}) + 5 (- 1)}

$\frac{(x - 6) (x - 1)}{5 (x^{2}) + 5 (- 1)}$

Step 2.1.3

Factor

5

$5$ out of

5 (x^{2}) + 5 (- 1)

$5 (x^{2}) + 5 (- 1)$ .

\frac{(x - 6) (x - 1)}{5 (x^{2} - 1)}

$\frac{(x - 6) (x - 1)}{5 (x^{2} - 1)}$

\frac{(x - 6) (x - 1)}{5 (x^{2} - 1)}

$\frac{(x - 6) (x - 1)}{5 (x^{2} - 1)}$

Step 2.2

Rewrite

1

$1$ as

1^{2}

$1^{2}$ .

\frac{(x - 6) (x - 1)}{5 (x^{2} - 1^{2})}

$\frac{(x - 6) (x - 1)}{5 (x^{2} - 1^{2})}$

Step 2.3

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = x

$a = x$ and

b = 1

$b = 1$ .

\frac{(x - 6) (x - 1)}{5 (x + 1) (x - 1)}

$\frac{(x - 6) (x - 1)}{5 (x + 1) (x - 1)}$

\frac{(x - 6) (x - 1)}{5 (x + 1) (x - 1)}

$\frac{(x - 6) (x - 1)}{5 (x + 1) (x - 1)}$

Step 3

Cancel the common factor of

x - 1

$x - 1$ .

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Step 3.1

Cancel the common factor.

$\frac{(x - 6) (x - 1)}{5 (x + 1) (x - 1)}$

Step 3.2

Rewrite the expression.

$\frac{x - 6}{5 (x + 1)}$