Algebra Examples

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

Step 1

Simplify terms.

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Step 1.1

Combine the numerators over the common denominator.

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

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Multiply $t$ by $t$ .

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

Step 1.2.3

Move $- 3$ to the left of $t$ .

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

Step 1.3

Subtract $t$ from $- 3 t$ .

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

Step 2

Factor using the perfect square rule.

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Step 2.1

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

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

Step 2.2

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

$4 t = 2 \cdot t \cdot 2$

Step 2.3

Rewrite the polynomial.

$\frac{t^{2} - 2 \cdot t \cdot 2 + 2^{2}}{t - 2}$

Step 2.4

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

$\frac{{(t - 2)}^{2}}{t - 2}$

Step 3

Cancel the common factor of ${(t - 2)}^{2}$ and $t - 2$ .

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Step 3.1

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

$\frac{(t - 2) (t - 2)}{t - 2}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Multiply by $1$ .

$\frac{(t - 2) (t - 2)}{(t - 2) \cdot 1}$

Step 3.2.2

Cancel the common factor.

$\frac{(t - 2) (t - 2)}{(t - 2) \cdot 1}$

Step 3.2.3

Rewrite the expression.

$\frac{t - 2}{1}$

Step 3.2.4

Divide $t - 2$ by $1$ .

$t - 2$