Basic Math Examples

$\frac{8}{t^{2} - 2 t - 3} + \frac{4}{t^{2} - 8 t + 15}$

Step 1

Simplify each term.

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Step 1.1

Factor $t^{2} - 2 t - 3$ using the AC method.

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Step 1.1.1

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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 1.1.2

Write the factored form using these integers.

$\frac{8}{(t - 3) (t + 1)} + \frac{4}{t^{2} - 8 t + 15}$

Step 1.2

Factor $t^{2} - 8 t + 15$ 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 $15$ and whose sum is $- 8$ .

$- 5, - 3$

Step 1.2.2

Write the factored form using these integers.

$\frac{8}{(t - 3) (t + 1)} + \frac{4}{(t - 5) (t - 3)}$

Step 2

To write $\frac{8}{(t - 3) (t + 1)}$ as a fraction with a common denominator, multiply by $\frac{t - 5}{t - 5}$ .

$\frac{8}{(t - 3) (t + 1)} \cdot \frac{t - 5}{t - 5} + \frac{4}{(t - 5) (t - 3)}$

Step 3

To write $\frac{4}{(t - 5) (t - 3)}$ as a fraction with a common denominator, multiply by $\frac{t + 1}{t + 1}$ .

$\frac{8}{(t - 3) (t + 1)} \cdot \frac{t - 5}{t - 5} + \frac{4}{(t - 5) (t - 3)} \cdot \frac{t + 1}{t + 1}$

Step 4

Write each expression with a common denominator of $(t - 3) (t + 1) (t - 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{8}{(t - 3) (t + 1)}$ by $\frac{t - 5}{t - 5}$ .

$\frac{8 (t - 5)}{(t - 3) (t + 1) (t - 5)} + \frac{4}{(t - 5) (t - 3)} \cdot \frac{t + 1}{t + 1}$

Step 4.2

Multiply $\frac{4}{(t - 5) (t - 3)}$ by $\frac{t + 1}{t + 1}$ .

$\frac{8 (t - 5)}{(t - 3) (t + 1) (t - 5)} + \frac{4 (t + 1)}{(t - 5) (t - 3) (t + 1)}$

Step 4.3

Reorder the factors of $(t - 3) (t + 1) (t - 5)$ .

$\frac{8 (t - 5)}{(t + 1) (t - 3) (t - 5)} + \frac{4 (t + 1)}{(t - 5) (t - 3) (t + 1)}$

Step 4.4

Reorder the factors of $(t - 5) (t - 3) (t + 1)$ .

$\frac{8 (t - 5)}{(t + 1) (t - 3) (t - 5)} + \frac{4 (t + 1)}{(t + 1) (t - 3) (t - 5)}$

Step 5

Combine the numerators over the common denominator.

$\frac{8 (t - 5) + 4 (t + 1)}{(t + 1) (t - 3) (t - 5)}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $4$ out of $8 (t - 5) + 4 (t + 1)$ .

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Step 6.1.1

Factor $4$ out of $8 (t - 5)$ .

$\frac{4 (2 (t - 5)) + 4 (t + 1)}{(t + 1) (t - 3) (t - 5)}$

Step 6.1.2

Factor $4$ out of $4 (2 (t - 5)) + 4 (t + 1)$ .

$\frac{4 (2 (t - 5) + t + 1)}{(t + 1) (t - 3) (t - 5)}$

Step 6.2

Apply the distributive property.

$\frac{4 (2 t + 2 \cdot - 5 + t + 1)}{(t + 1) (t - 3) (t - 5)}$

Step 6.3

Multiply $2$ by $- 5$ .

$\frac{4 (2 t - 10 + t + 1)}{(t + 1) (t - 3) (t - 5)}$

Step 6.4

Add $2 t$ and $t$ .

$\frac{4 (3 t - 10 + 1)}{(t + 1) (t - 3) (t - 5)}$

Step 6.5

Add $- 10$ and $1$ .

$\frac{4 (3 t - 9)}{(t + 1) (t - 3) (t - 5)}$

Step 6.6

Factor $3$ out of $3 t - 9$ .

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Step 6.6.1

Factor $3$ out of $3 t$ .

$\frac{4 (3 (t) - 9)}{(t + 1) (t - 3) (t - 5)}$

Step 6.6.2

Factor $3$ out of $- 9$ .

$\frac{4 (3 t + 3 \cdot - 3)}{(t + 1) (t - 3) (t - 5)}$

Step 6.6.3

Factor $3$ out of $3 t + 3 \cdot - 3$ .

$\frac{4 (3 (t - 3))}{(t + 1) (t - 3) (t - 5)}$

$\frac{4 \cdot 3 (t - 3)}{(t + 1) (t - 3) (t - 5)}$

Step 6.7

Multiply $4$ by $3$ .

$\frac{12 (t - 3)}{(t + 1) (t - 3) (t - 5)}$

Step 7

Cancel the common factor of $t - 3$ .

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Step 7.1

Cancel the common factor.

$\frac{12 (t - 3)}{(t + 1) (t - 3) (t - 5)}$

Step 7.2

Rewrite the expression.

$\frac{12}{(t + 1) (t - 5)}$