Basic Math Examples

$\frac{7 + 6 t - t^{2}}{t^{2} + 12 t + 11}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $7$ as $16$ plus $- 9$

$\frac{16 - 9 + 6 t - t^{2}}{t^{2} + 12 t + 11}$

Step 1.2

Factor using the perfect square rule.

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Step 1.2.1

Rewrite $9$ as $3^{2}$ .

$\frac{16 - (3^{2} - 6 t + t^{2})}{t^{2} + 12 t + 11}$

Step 1.2.2

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

$6 t = 2 \cdot 3 \cdot t$

Step 1.2.3

Rewrite the polynomial.

$\frac{16 - (3^{2} - 2 \cdot 3 \cdot t + t^{2})}{t^{2} + 12 t + 11}$

Step 1.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 3$ and $b = t$ .

$\frac{16 - {(3 - t)}^{2}}{t^{2} + 12 t + 11}$

Step 1.3

Rewrite $16$ as $4^{2}$ .

$\frac{4^{2} - {(3 - t)}^{2}}{t^{2} + 12 t + 11}$

Step 1.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4$ and $b = 3 - t$ .

$\frac{(4 + 3 - t) (4 - (3 - t))}{t^{2} + 12 t + 11}$

Step 1.5

Simplify.

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Step 1.5.1

Add $4$ and $3$ .

$\frac{(7 - t) (4 - (3 - t))}{t^{2} + 12 t + 11}$

Step 1.5.2

Apply the distributive property.

$\frac{(7 - t) (4 - 1 \cdot 3 - - t)}{t^{2} + 12 t + 11}$

Step 1.5.3

Multiply $- 1$ by $3$ .

$\frac{(7 - t) (4 - 3 - - t)}{t^{2} + 12 t + 11}$

Step 1.5.4

Multiply $- - t$ .

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Step 1.5.4.1

Multiply $- 1$ by $- 1$ .

$\frac{(7 - t) (4 - 3 + 1 t)}{t^{2} + 12 t + 11}$

Step 1.5.4.2

Multiply $t$ by $1$ .

$\frac{(7 - t) (4 - 3 + t)}{t^{2} + 12 t + 11}$

Step 1.5.5

Subtract $3$ from $4$ .

$\frac{(7 - t) (1 + t)}{t^{2} + 12 t + 11}$

Step 2

Factor $t^{2} + 12 t + 11$ 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 $11$ and whose sum is $12$ .

$1, 11$

Step 2.2

Write the factored form using these integers.

$\frac{(7 - t) (1 + t)}{(t + 1) (t + 11)}$

Step 3

Cancel the common factor of $1 + t$ and $t + 1$ .

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Step 3.1

Reorder terms.

$\frac{(7 - t) (t + 1)}{(t + 1) (t + 11)}$

Step 3.2

Cancel the common factor.

$\frac{(7 - t) (t + 1)}{(t + 1) (t + 11)}$

Step 3.3

Rewrite the expression.

$\frac{7 - t}{t + 11}$