Basic Math Examples

$\frac{z^{2} + 8 z + 7}{6 z^{2} - 6}$

Step 1

Factor $z^{2} + 8 z + 7$ using the AC method.

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Step 1.1

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 $7$ and whose sum is $8$ .

$1, 7$

Step 1.2

Write the factored form using these integers.

$\frac{(z + 1) (z + 7)}{6 z^{2} - 6}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $6$ out of $6 z^{2} - 6$ .

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Step 2.1.1

Factor $6$ out of $6 z^{2}$ .

$\frac{(z + 1) (z + 7)}{6 (z^{2}) - 6}$

Step 2.1.2

Factor $6$ out of $- 6$ .

$\frac{(z + 1) (z + 7)}{6 (z^{2}) + 6 (- 1)}$

Step 2.1.3

Factor $6$ out of $6 (z^{2}) + 6 (- 1)$ .

$\frac{(z + 1) (z + 7)}{6 (z^{2} - 1)}$

Step 2.2

Rewrite $1$ as $1^{2}$ .

$\frac{(z + 1) (z + 7)}{6 (z^{2} - 1^{2})}$

Step 2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = z$ and $b = 1$ .

$\frac{(z + 1) (z + 7)}{6 (z + 1) (z - 1)}$

Step 3

Cancel the common factor of $z + 1$ .

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Step 3.1

Cancel the common factor.

$\frac{(z + 1) (z + 7)}{6 (z + 1) (z - 1)}$

Step 3.2

Rewrite the expression.

$\frac{z + 7}{6 (z - 1)}$