Algebra Examples

$\frac{3 a^{2} + a - 1}{a^{2} - 2 a + 1} - \frac{2 a^{2} - a + 2}{a^{2} - 2 a + 1}$

Step 1

Simplify terms.

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Step 1.1

Combine the numerators over the common denominator.

$\frac{3 a^{2} + a - 1 - (2 a^{2} - a + 2)}{a^{2} - 2 a + 1}$

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the distributive property.

$\frac{3 a^{2} + a - 1 - (2 a^{2}) - - a - 1 \cdot 2}{a^{2} - 2 a + 1}$

Step 1.2.2

Simplify.

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Step 1.2.2.1

Multiply $2$ by $- 1$ .

$\frac{3 a^{2} + a - 1 - 2 a^{2} - - a - 1 \cdot 2}{a^{2} - 2 a + 1}$

Step 1.2.2.2

Multiply $- - a$ .

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Step 1.2.2.2.1

Multiply $- 1$ by $- 1$ .

$\frac{3 a^{2} + a - 1 - 2 a^{2} + 1 a - 1 \cdot 2}{a^{2} - 2 a + 1}$

Step 1.2.2.2.2

Multiply $a$ by $1$ .

$\frac{3 a^{2} + a - 1 - 2 a^{2} + a - 1 \cdot 2}{a^{2} - 2 a + 1}$

Step 1.2.2.3

Multiply $- 1$ by $2$ .

$\frac{3 a^{2} + a - 1 - 2 a^{2} + a - 2}{a^{2} - 2 a + 1}$

Step 1.3

Simplify by adding terms.

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Step 1.3.1

Subtract $2 a^{2}$ from $3 a^{2}$ .

$\frac{a^{2} + a - 1 + a - 2}{a^{2} - 2 a + 1}$

Step 1.3.2

Add $a$ and $a$ .

$\frac{a^{2} + 2 a - 1 - 2}{a^{2} - 2 a + 1}$

Step 1.3.3

Subtract $2$ from $- 1$ .

$\frac{a^{2} + 2 a - 3}{a^{2} - 2 a + 1}$

Step 2

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

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

$- 1, 3$

Step 2.2

Write the factored form using these integers.

$\frac{(a - 1) (a + 3)}{a^{2} - 2 a + 1}$

Step 3

Factor using the perfect square rule.

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Step 3.1

Rewrite $1$ as $1^{2}$ .

$\frac{(a - 1) (a + 3)}{a^{2} - 2 a + 1^{2}}$

Step 3.2

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

$2 a = 2 \cdot a \cdot 1$

Step 3.3

Rewrite the polynomial.

$\frac{(a - 1) (a + 3)}{a^{2} - 2 \cdot a \cdot 1 + 1^{2}}$

Step 3.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = a$ and $b = 1$ .

$\frac{(a - 1) (a + 3)}{{(a - 1)}^{2}}$

Step 4

Cancel the common factors.

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Step 4.1

Factor $a - 1$ out of ${(a - 1)}^{2}$ .

$\frac{(a - 1) (a + 3)}{(a - 1) (a - 1)}$

Step 4.2

Cancel the common factor.

$\frac{(a - 1) (a + 3)}{(a - 1) (a - 1)}$

Step 4.3

Rewrite the expression.

$\frac{a + 3}{a - 1}$