Basic Math Examples

$\frac{t}{t^{2} + 14 t + 49} + \frac{7}{t^{2} + 13 t + 42}$

Step 1

Simplify each term.

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Step 1.1

Factor using the perfect square rule.

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Step 1.1.1

Rewrite $49$ as $7^{2}$ .

$\frac{t}{t^{2} + 14 t + 7^{2}} + \frac{7}{t^{2} + 13 t + 42}$

Step 1.1.2

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

$14 t = 2 \cdot t \cdot 7$

Step 1.1.3

Rewrite the polynomial.

$\frac{t}{t^{2} + 2 \cdot t \cdot 7 + 7^{2}} + \frac{7}{t^{2} + 13 t + 42}$

Step 1.1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = t$ and $b = 7$ .

$\frac{t}{{(t + 7)}^{2}} + \frac{7}{t^{2} + 13 t + 42}$

Step 1.2

Factor $t^{2} + 13 t + 42$ using the AC method.

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Step 1.2.1

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 $42$ and whose sum is $13$ .

$6, 7$

Step 1.2.2

Write the factored form using these integers.

$\frac{t}{{(t + 7)}^{2}} + \frac{7}{(t + 6) (t + 7)}$

Step 2

To write $\frac{t}{{(t + 7)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{t + 6}{t + 6}$ .

$\frac{t}{{(t + 7)}^{2}} \cdot \frac{t + 6}{t + 6} + \frac{7}{(t + 6) (t + 7)}$

Step 3

To write $\frac{7}{(t + 6) (t + 7)}$ as a fraction with a common denominator, multiply by $\frac{t + 7}{t + 7}$ .

$\frac{t}{{(t + 7)}^{2}} \cdot \frac{t + 6}{t + 6} + \frac{7}{(t + 6) (t + 7)} \cdot \frac{t + 7}{t + 7}$

Step 4

Write each expression with a common denominator of $(t + 6) {(t + 7)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{t}{{(t + 7)}^{2}}$ by $\frac{t + 6}{t + 6}$ .

$\frac{t (t + 6)}{{(t + 7)}^{2} (t + 6)} + \frac{7}{(t + 6) (t + 7)} \cdot \frac{t + 7}{t + 7}$

Step 4.2

Multiply $\frac{7}{(t + 6) (t + 7)}$ by $\frac{t + 7}{t + 7}$ .

$\frac{t (t + 6)}{{(t + 7)}^{2} (t + 6)} + \frac{7 (t + 7)}{(t + 6) (t + 7) (t + 7)}$

Step 4.3

Raise $t + 7$ to the power of $1$ .

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

Step 4.4

Raise $t + 7$ to the power of $1$ .

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

Step 4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.6

Add $1$ and $1$ .

$\frac{t (t + 6)}{{(t + 7)}^{2} (t + 6)} + \frac{7 (t + 7)}{(t + 6) {(t + 7)}^{2}}$

Step 4.7

Reorder the factors of $(t + 6) {(t + 7)}^{2}$ .

$\frac{t (t + 6)}{{(t + 7)}^{2} (t + 6)} + \frac{7 (t + 7)}{{(t + 7)}^{2} (t + 6)}$

Step 5

Combine the numerators over the common denominator.

$\frac{t (t + 6) + 7 (t + 7)}{{(t + 7)}^{2} (t + 6)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{t \cdot t + t \cdot 6 + 7 (t + 7)}{{(t + 7)}^{2} (t + 6)}$

Step 6.2

Multiply $t$ by $t$ .

$\frac{t^{2} + t \cdot 6 + 7 (t + 7)}{{(t + 7)}^{2} (t + 6)}$

Step 6.3

Move $6$ to the left of $t$ .

$\frac{t^{2} + 6 \cdot t + 7 (t + 7)}{{(t + 7)}^{2} (t + 6)}$

Step 6.4

Apply the distributive property.

$\frac{t^{2} + 6 t + 7 t + 7 \cdot 7}{{(t + 7)}^{2} (t + 6)}$

Step 6.5

Multiply $7$ by $7$ .

$\frac{t^{2} + 6 t + 7 t + 49}{{(t + 7)}^{2} (t + 6)}$

Step 6.6

Add $6 t$ and $7 t$ .

$\frac{t^{2} + 13 t + 49}{{(t + 7)}^{2} (t + 6)}$