Algebra Examples

$\frac{t - 3}{5 - t} - \frac{t + 4}{t + 5} + \frac{t - 2}{25 - t^{2}}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $25$ as $5^{2}$ .

$\frac{t - 3}{5 - t} - \frac{t + 4}{t + 5} + \frac{t - 2}{5^{2} - t^{2}}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5$ and $b = t$ .

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

Multiply $\frac{t - 3}{5 - t}$ by $\frac{t + 5}{t + 5}$ .

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

Step 4.2

Multiply $\frac{t + 4}{t + 5}$ by $\frac{5 - t}{5 - t}$ .

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

Step 4.3

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Expand $(t - 3) (t + 5)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

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

Step 6.1.2

Apply the distributive property.

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

Step 6.1.3

Apply the distributive property.

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

Step 6.2

Simplify and combine like terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $t$ by $t$ .

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

Step 6.2.1.2

Move $5$ to the left of $t$ .

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

Step 6.2.1.3

Multiply $- 3$ by $5$ .

$\frac{t^{2} + 5 t - 3 t - 15 - (t + 4) (5 - t)}{(5 - t) (t + 5)} + \frac{t - 2}{(5 + t) (5 - t)}$

Step 6.2.2

Subtract $3 t$ from $5 t$ .

$\frac{t^{2} + 2 t - 15 - (t + 4) (5 - t)}{(5 - t) (t + 5)} + \frac{t - 2}{(5 + t) (5 - t)}$

Step 6.3

Apply the distributive property.

$\frac{t^{2} + 2 t - 15 + (- t - 1 \cdot 4) (5 - t)}{(5 - t) (t + 5)} + \frac{t - 2}{(5 + t) (5 - t)}$

Step 6.4

Multiply $- 1$ by $4$ .

$\frac{t^{2} + 2 t - 15 + (- t - 4) (5 - t)}{(5 - t) (t + 5)} + \frac{t - 2}{(5 + t) (5 - t)}$

Step 6.5

Expand $(- t - 4) (5 - t)$ using the FOIL Method.

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Step 6.5.1

Apply the distributive property.

$\frac{t^{2} + 2 t - 15 - t (5 - t) - 4 (5 - t)}{(5 - t) (t + 5)} + \frac{t - 2}{(5 + t) (5 - t)}$

Step 6.5.2

Apply the distributive property.

$\frac{t^{2} + 2 t - 15 - t \cdot 5 - t (- t) - 4 (5 - t)}{(5 - t) (t + 5)} + \frac{t - 2}{(5 + t) (5 - t)}$

Step 6.5.3

Apply the distributive property.

$\frac{t^{2} + 2 t - 15 - t \cdot 5 - t (- t) - 4 \cdot 5 - 4 (- t)}{(5 - t) (t + 5)} + \frac{t - 2}{(5 + t) (5 - t)}$

Step 6.6

Simplify and combine like terms.

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Step 6.6.1

Simplify each term.

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Step 6.6.1.1

Multiply $5$ by $- 1$ .

$\frac{t^{2} + 2 t - 15 - 5 t - t (- t) - 4 \cdot 5 - 4 (- t)}{(5 - t) (t + 5)} + \frac{t - 2}{(5 + t) (5 - t)}$

Step 6.6.1.2

Rewrite using the commutative property of multiplication.

$\frac{t^{2} + 2 t - 15 - 5 t - 1 \cdot - 1 t \cdot t - 4 \cdot 5 - 4 (- t)}{(5 - t) (t + 5)} + \frac{t - 2}{(5 + t) (5 - t)}$

Step 6.6.1.3

Multiply $t$ by $t$ by adding the exponents.

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Step 6.6.1.3.1

Move $t$ .

$\frac{t^{2} + 2 t - 15 - 5 t - 1 \cdot - 1 (t \cdot t) - 4 \cdot 5 - 4 (- t)}{(5 - t) (t + 5)} + \frac{t - 2}{(5 + t) (5 - t)}$

Step 6.6.1.3.2

Multiply $t$ by $t$ .

$\frac{t^{2} + 2 t - 15 - 5 t - 1 \cdot - 1 t^{2} - 4 \cdot 5 - 4 (- t)}{(5 - t) (t + 5)} + \frac{t - 2}{(5 + t) (5 - t)}$

Step 6.6.1.4

Multiply $- 1$ by $- 1$ .

$\frac{t^{2} + 2 t - 15 - 5 t + 1 t^{2} - 4 \cdot 5 - 4 (- t)}{(5 - t) (t + 5)} + \frac{t - 2}{(5 + t) (5 - t)}$

Step 6.6.1.5

Multiply $t^{2}$ by $1$ .

$\frac{t^{2} + 2 t - 15 - 5 t + t^{2} - 4 \cdot 5 - 4 (- t)}{(5 - t) (t + 5)} + \frac{t - 2}{(5 + t) (5 - t)}$

Step 6.6.1.6

Multiply $- 4$ by $5$ .

$\frac{t^{2} + 2 t - 15 - 5 t + t^{2} - 20 - 4 (- t)}{(5 - t) (t + 5)} + \frac{t - 2}{(5 + t) (5 - t)}$

Step 6.6.1.7

Multiply $- 1$ by $- 4$ .

$\frac{t^{2} + 2 t - 15 - 5 t + t^{2} - 20 + 4 t}{(5 - t) (t + 5)} + \frac{t - 2}{(5 + t) (5 - t)}$

Step 6.6.2

Add $- 5 t$ and $4 t$ .

$\frac{t^{2} + 2 t - 15 - t + t^{2} - 20}{(5 - t) (t + 5)} + \frac{t - 2}{(5 + t) (5 - t)}$

Step 6.7

Add $t^{2}$ and $t^{2}$ .

$\frac{2 t^{2} + 2 t - 15 - t - 20}{(5 - t) (t + 5)} + \frac{t - 2}{(5 + t) (5 - t)}$

Step 6.8

Subtract $t$ from $2 t$ .

$\frac{2 t^{2} + t - 15 - 20}{(5 - t) (t + 5)} + \frac{t - 2}{(5 + t) (5 - t)}$

Step 6.9

Subtract $20$ from $- 15$ .

$\frac{2 t^{2} + t - 35}{(5 - t) (t + 5)} + \frac{t - 2}{(5 + t) (5 - t)}$

Step 7

Combine into one fraction.

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Step 7.1

Reorder terms.

$\frac{2 t^{2} + t - 35}{(5 - t) (t + 5)} + \frac{t - 2}{(t + 5) (5 - t)}$

Step 7.2

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

$\frac{2 t^{2} + t - 35}{(5 - t) (t + 5)} + \frac{t - 2}{(5 - t) (t + 5)}$

Step 7.3

Combine the numerators over the common denominator.

$\frac{2 t^{2} + t - 35 + t - 2}{(5 - t) (t + 5)}$

Step 8

Simplify the numerator.

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Step 8.1

Add $t$ and $t$ .

$\frac{2 t^{2} + 2 t - 35 - 2}{(5 - t) (t + 5)}$

Step 8.2

Subtract $2$ from $- 35$ .

$\frac{2 t^{2} + 2 t - 37}{(5 - t) (t + 5)}$