Algebra Examples

$\frac{5}{t^{2}} - \frac{4}{t + 1}$

Step 1

To write $\frac{5}{t^{2}}$ as a fraction with a common denominator, multiply by $\frac{t + 1}{t + 1}$ .

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

Step 2

To write $- \frac{4}{t + 1}$ as a fraction with a common denominator, multiply by $\frac{t^{2}}{t^{2}}$ .

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

Step 3

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

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Step 3.1

Multiply $\frac{5}{t^{2}}$ by $\frac{t + 1}{t + 1}$ .

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

Step 3.2

Multiply $\frac{4}{t + 1}$ by $\frac{t^{2}}{t^{2}}$ .

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

Step 3.3

Reorder the factors of $(t + 1) t^{2}$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $5$ by $1$ .

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

Step 5.3

Reorder terms.

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

Step 6

Simplify with factoring out.

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Step 6.1

Factor $- 1$ out of $- 4 t^{2}$ .

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

Step 6.2

Factor $- 1$ out of $5 t$ .

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

Step 6.3

Factor $- 1$ out of $- (4 t^{2}) - (- 5 t)$ .

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

Step 6.4

Rewrite $5$ as $- 1 (- 5)$ .

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

Step 6.5

Factor $- 1$ out of $- (4 t^{2} - 5 t) - 1 (- 5)$ .

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

Step 6.6

Simplify the expression.

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Step 6.6.1

Rewrite $- (4 t^{2} - 5 t - 5)$ as $- 1 (4 t^{2} - 5 t - 5)$ .

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

Step 6.6.2

Move the negative in front of the fraction.

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