Basic Math Examples

$\frac{z^{2} + 10 z + 24}{z^{2} + 11 z + 28} \div \frac{z^{2} + 6 z}{z^{2} - z - 56}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{z^{2} + 10 z + 24}{z^{2} + 11 z + 28} \cdot \frac{z^{2} - z - 56}{z^{2} + 6 z}$

Step 2

Factor $z^{2} + 10 z + 24$ 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 $24$ and whose sum is $10$ .

$4, 6$

Step 2.2

Write the factored form using these integers.

$\frac{(z + 4) (z + 6)}{z^{2} + 11 z + 28} \cdot \frac{z^{2} - z - 56}{z^{2} + 6 z}$

Step 3

Factor $z^{2} + 11 z + 28$ 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 $28$ and whose sum is $11$ .

$4, 7$

Step 3.2

Write the factored form using these integers.

$\frac{(z + 4) (z + 6)}{(z + 4) (z + 7)} \cdot \frac{z^{2} - z - 56}{z^{2} + 6 z}$

Step 4

Cancel the common factor of $z + 4$ .

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Step 4.1

Cancel the common factor.

$\frac{(z + 4) (z + 6)}{(z + 4) (z + 7)} \cdot \frac{z^{2} - z - 56}{z^{2} + 6 z}$

Step 4.2

Rewrite the expression.

$\frac{z + 6}{z + 7} \cdot \frac{z^{2} - z - 56}{z^{2} + 6 z}$

Step 5

Factor $z^{2} - z - 56$ using the AC method.

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Step 5.1

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 $- 56$ and whose sum is $- 1$ .

$- 8, 7$

Step 5.2

Write the factored form using these integers.

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

Step 6

Simplify terms.

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Step 6.1

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

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Step 6.1.1

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

$\frac{z + 6}{z + 7} \cdot \frac{(z - 8) (z + 7)}{z \cdot z + 6 z}$

Step 6.1.2

Factor $z$ out of $6 z$ .

$\frac{z + 6}{z + 7} \cdot \frac{(z - 8) (z + 7)}{z \cdot z + z \cdot 6}$

Step 6.1.3

Factor $z$ out of $z \cdot z + z \cdot 6$ .

$\frac{z + 6}{z + 7} \cdot \frac{(z - 8) (z + 7)}{z (z + 6)}$

Step 6.2

Cancel the common factor of $z + 6$ .

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Step 6.2.1

Factor $z + 6$ out of $z (z + 6)$ .

$\frac{z + 6}{z + 7} \cdot \frac{(z - 8) (z + 7)}{(z + 6) z}$

Step 6.2.2

Cancel the common factor.

$\frac{z + 6}{z + 7} \cdot \frac{(z - 8) (z + 7)}{(z + 6) z}$

Step 6.2.3

Rewrite the expression.

$\frac{1}{z + 7} \cdot \frac{(z - 8) (z + 7)}{z}$

Step 6.3

Cancel the common factor of $z + 7$ .

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Step 6.3.1

Factor $z + 7$ out of $(z - 8) (z + 7)$ .

$\frac{1}{z + 7} \cdot \frac{(z + 7) (z - 8)}{z}$

Step 6.3.2

Cancel the common factor.

$\frac{1}{z + 7} \cdot \frac{(z + 7) (z - 8)}{z}$

Step 6.3.3

Rewrite the expression.

$\frac{z - 8}{z}$