Algebra Examples

$\frac{(x^{2} - 3 x^{2} + 2 x - 1) d x}{x^{2} - 4 x + 4}$

Step 1

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

$\frac{(- 2 x^{2} + 2 x - 1) d x}{x^{2} - 4 x + 4}$

Step 2

Factor using the perfect square rule.

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Step 2.1

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

$\frac{(- 2 x^{2} + 2 x - 1) d x}{x^{2} - 4 x + 2^{2}}$

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.

$4 x = 2 \cdot x \cdot 2$

Step 2.3

Rewrite the polynomial.

$\frac{(- 2 x^{2} + 2 x - 1) d x}{x^{2} - 2 \cdot x \cdot 2 + 2^{2}}$

Step 2.4

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

$\frac{(- 2 x^{2} + 2 x - 1) d x}{{(x - 2)}^{2}}$

Step 3

Simplify with factoring out.

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Step 3.1

Factor $- 1$ out of $- 2 x^{2}$ .

$\frac{(- (2 x^{2}) + 2 x - 1) d x}{{(x - 2)}^{2}}$

Step 3.2

Factor $- 1$ out of $2 x$ .

$\frac{(- (2 x^{2}) - (- 2 x) - 1) d x}{{(x - 2)}^{2}}$

Step 3.3

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

$\frac{(- (2 x^{2} - 2 x) - 1) d x}{{(x - 2)}^{2}}$

Step 3.4

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

$\frac{(- (2 x^{2} - 2 x) - 1 (1)) d x}{{(x - 2)}^{2}}$

Step 3.5

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

$\frac{- (2 x^{2} - 2 x + 1) d x}{{(x - 2)}^{2}}$

Step 3.6

Simplify the expression.

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Step 3.6.1

Rewrite $- (2 x^{2} - 2 x + 1)$ as $- 1 (2 x^{2} - 2 x + 1)$ .

$\frac{- 1 (2 x^{2} - 2 x + 1) d x}{{(x - 2)}^{2}}$

Step 3.6.2

Move the negative in front of the fraction.

$- \frac{(2 x^{2} - 2 x + 1) d x}{{(x - 2)}^{2}}$

Step 3.6.3

Reorder factors in $- \frac{(2 x^{2} - 2 x + 1) d x}{{(x - 2)}^{2}}$ .

$- \frac{d x (2 x^{2} - 2 x + 1)}{{(x - 2)}^{2}}$