Basic Math Examples

$\frac{z^{2} - 3 z}{z^{2} + 4 z + 4} - \frac{10}{z^{2} + 4 z + 4}$

Step 1

Combine the numerators over the common denominator.

$\frac{z^{2} - 3 z - 10}{z^{2} + 4 z + 4}$

Step 2

Factor $z^{2} - 3 z - 10$ 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 $- 10$ and whose sum is $- 3$ .

$- 5, 2$

Step 2.2

Write the factored form using these integers.

$\frac{(z - 5) (z + 2)}{z^{2} + 4 z + 4}$

Step 3

Factor using the perfect square rule.

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Step 3.1

Rewrite $4$ as $2^{2}$ .

$\frac{(z - 5) (z + 2)}{z^{2} + 4 z + 2^{2}}$

Step 3.2

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

$4 z = 2 \cdot z \cdot 2$

Step 3.3

Rewrite the polynomial.

$\frac{(z - 5) (z + 2)}{z^{2} + 2 \cdot z \cdot 2 + 2^{2}}$

Step 3.4

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

$\frac{(z - 5) (z + 2)}{{(z + 2)}^{2}}$

Step 4

Cancel the common factor of $z + 2$ and ${(z + 2)}^{2}$ .

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Step 4.1

Factor $z + 2$ out of $(z - 5) (z + 2)$ .

$\frac{(z + 2) (z - 5)}{{(z + 2)}^{2}}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

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

$\frac{(z + 2) (z - 5)}{(z + 2) (z + 2)}$

Step 4.2.2

Cancel the common factor.

$\frac{(z + 2) (z - 5)}{(z + 2) (z + 2)}$

Step 4.2.3

Rewrite the expression.

$\frac{z - 5}{z + 2}$