Pre-Algebra Examples

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

Step 1

Simplify $\frac{t}{t - 2} + \frac{4}{t + 2}$ .

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Step 1.1

To write $\frac{t}{t - 2}$ as a fraction with a common denominator, multiply by $\frac{t + 2}{t + 2}$ .

$\frac{t}{t - 2} \cdot \frac{t + 2}{t + 2} + \frac{4}{t + 2} = \frac{8}{t^{2} - 4}$

Step 1.2

To write $\frac{4}{t + 2}$ as a fraction with a common denominator, multiply by $\frac{t - 2}{t - 2}$ .

$\frac{t}{t - 2} \cdot \frac{t + 2}{t + 2} + \frac{4}{t + 2} \cdot \frac{t - 2}{t - 2} = \frac{8}{t^{2} - 4}$

Step 1.3

Write each expression with a common denominator of $(t - 2) (t + 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{t}{t - 2}$ by $\frac{t + 2}{t + 2}$ .

$\frac{t (t + 2)}{(t - 2) (t + 2)} + \frac{4}{t + 2} \cdot \frac{t - 2}{t - 2} = \frac{8}{t^{2} - 4}$

Step 1.3.2

Multiply $\frac{4}{t + 2}$ by $\frac{t - 2}{t - 2}$ .

$\frac{t (t + 2)}{(t - 2) (t + 2)} + \frac{4 (t - 2)}{(t + 2) (t - 2)} = \frac{8}{t^{2} - 4}$

Step 1.3.3

Reorder the factors of $(t - 2) (t + 2)$ .

$\frac{t (t + 2)}{(t + 2) (t - 2)} + \frac{4 (t - 2)}{(t + 2) (t - 2)} = \frac{8}{t^{2} - 4}$

Step 1.4

Combine the numerators over the common denominator.

$\frac{t (t + 2) + 4 (t - 2)}{(t + 2) (t - 2)} = \frac{8}{t^{2} - 4}$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Apply the distributive property.

$\frac{t \cdot t + t \cdot 2 + 4 (t - 2)}{(t + 2) (t - 2)} = \frac{8}{t^{2} - 4}$

Step 1.5.2

Multiply $t$ by $t$ .

$\frac{t^{2} + t \cdot 2 + 4 (t - 2)}{(t + 2) (t - 2)} = \frac{8}{t^{2} - 4}$

Step 1.5.3

Move $2$ to the left of $t$ .

$\frac{t^{2} + 2 \cdot t + 4 (t - 2)}{(t + 2) (t - 2)} = \frac{8}{t^{2} - 4}$

Step 1.5.4

Apply the distributive property.

$\frac{t^{2} + 2 t + 4 t + 4 \cdot - 2}{(t + 2) (t - 2)} = \frac{8}{t^{2} - 4}$

Step 1.5.5

Multiply $4$ by $- 2$ .

$\frac{t^{2} + 2 t + 4 t - 8}{(t + 2) (t - 2)} = \frac{8}{t^{2} - 4}$

Step 1.5.6

Add $2 t$ and $4 t$ .

$\frac{t^{2} + 6 t - 8}{(t + 2) (t - 2)} = \frac{8}{t^{2} - 4}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{t^{2} + 6 t - 8}{(t + 2) (t - 2)} = \frac{8}{t^{2} - 2^{2}}$

Step 2.2

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

$\frac{t^{2} + 6 t - 8}{(t + 2) (t - 2)} = \frac{8}{(t + 2) (t - 2)}$

Step 3

Graph each side of the equation. The solution is the x-value of the point of intersection.

$t = - 8$

Step 4