Basic Math Examples

$\frac{4 t^{2} + 16 t + 16}{16 t^{2} - 16 t - 96}$

Step 1

Cancel the common factor of $4 t^{2} + 16 t + 16$ and $16 t^{2} - 16 t - 96$ .

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Step 1.1

Factor $4$ out of $4 t^{2}$ .

$\frac{4 (t^{2}) + 16 t + 16}{16 t^{2} - 16 t - 96}$

Step 1.2

Factor $4$ out of $16 t$ .

$\frac{4 (t^{2}) + 4 (4 t) + 16}{16 t^{2} - 16 t - 96}$

Step 1.3

Factor $4$ out of $4 (t^{2}) + 4 (4 t)$ .

$\frac{4 (t^{2} + 4 t) + 16}{16 t^{2} - 16 t - 96}$

Step 1.4

Factor $4$ out of $16$ .

$\frac{4 (t^{2} + 4 t) + 4 \cdot 4}{16 t^{2} - 16 t - 96}$

Step 1.5

Factor $4$ out of $4 (t^{2} + 4 t) + 4 \cdot 4$ .

$\frac{4 (t^{2} + 4 t + 4)}{16 t^{2} - 16 t - 96}$

Step 1.6

Cancel the common factors.

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Step 1.6.1

Factor $4$ out of $16 t^{2}$ .

$\frac{4 (t^{2} + 4 t + 4)}{4 (4 t^{2}) - 16 t - 96}$

Step 1.6.2

Factor $4$ out of $- 16 t$ .

$\frac{4 (t^{2} + 4 t + 4)}{4 (4 t^{2}) + 4 (- 4 t) - 96}$

Step 1.6.3

Factor $4$ out of $4 (4 t^{2}) + 4 (- 4 t)$ .

$\frac{4 (t^{2} + 4 t + 4)}{4 (4 t^{2} - 4 t) - 96}$

Step 1.6.4

Factor $4$ out of $- 96$ .

$\frac{4 (t^{2} + 4 t + 4)}{4 (4 t^{2} - 4 t) + 4 (- 24)}$

Step 1.6.5

Factor $4$ out of $4 (4 t^{2} - 4 t) + 4 (- 24)$ .

$\frac{4 (t^{2} + 4 t + 4)}{4 (4 t^{2} - 4 t - 24)}$

Step 1.6.6

Cancel the common factor.

$\frac{4 (t^{2} + 4 t + 4)}{4 (4 t^{2} - 4 t - 24)}$

Step 1.6.7

Rewrite the expression.

$\frac{t^{2} + 4 t + 4}{4 t^{2} - 4 t - 24}$

Step 2

Factor using the perfect square rule.

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Step 2.1

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

$\frac{t^{2} + 4 t + 2^{2}}{4 t^{2} - 4 t - 24}$

Step 2.2

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

$4 t = 2 \cdot t \cdot 2$

Step 2.3

Rewrite the polynomial.

$\frac{t^{2} + 2 \cdot t \cdot 2 + 2^{2}}{4 t^{2} - 4 t - 24}$

Step 2.4

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

$\frac{{(t + 2)}^{2}}{4 t^{2} - 4 t - 24}$

Step 3

Simplify the denominator.

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Step 3.1

Factor $4$ out of $4 t^{2} - 4 t - 24$ .

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Step 3.1.1

Factor $4$ out of $4 t^{2}$ .

$\frac{{(t + 2)}^{2}}{4 (t^{2}) - 4 t - 24}$

Step 3.1.2

Factor $4$ out of $- 4 t$ .

$\frac{{(t + 2)}^{2}}{4 (t^{2}) + 4 (- t) - 24}$

Step 3.1.3

Factor $4$ out of $- 24$ .

$\frac{{(t + 2)}^{2}}{4 (t^{2}) + 4 (- t) + 4 (- 6)}$

Step 3.1.4

Factor $4$ out of $4 (t^{2}) + 4 (- t)$ .

$\frac{{(t + 2)}^{2}}{4 (t^{2} - t) + 4 (- 6)}$

Step 3.1.5

Factor $4$ out of $4 (t^{2} - t) + 4 (- 6)$ .

$\frac{{(t + 2)}^{2}}{4 (t^{2} - t - 6)}$

Step 3.2

Factor $t^{2} - t - 6$ using the AC method.

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Step 3.2.1

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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 3.2.2

Write the factored form using these integers.

$\frac{{(t + 2)}^{2}}{4 ((t - 3) (t + 2))}$

$\frac{{(t + 2)}^{2}}{4 (t - 3) (t + 2)}$

Step 4

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

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Step 4.1

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

$\frac{(t + 2) (t + 2)}{4 (t - 3) (t + 2)}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $t + 2$ out of $4 (t - 3) (t + 2)$ .

$\frac{(t + 2) (t + 2)}{(t + 2) (4 (t - 3))}$

Step 4.2.2

Cancel the common factor.

$\frac{(t + 2) (t + 2)}{(t + 2) (4 (t - 3))}$

Step 4.2.3

Rewrite the expression.

$\frac{t + 2}{4 (t - 3)}$