Basic Math Examples

$a^{2} - b^{2} - 2 a + 1$

Step 1

Regroup terms.

$- b^{2} + a^{2} - 2 a + 1$

Step 2

Factor using the perfect square rule.

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Step 2.1

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

$- b^{2} + a^{2} - 2 a + 1^{2}$

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.

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

Step 2.3

Rewrite the polynomial.

$- b^{2} + a^{2} - 2 \cdot a \cdot 1 + 1^{2}$

Step 2.4

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

$- b^{2} + {(a - 1)}^{2}$

Step 3

Reorder $- b^{2}$ and ${(a - 1)}^{2}$ .

${(a - 1)}^{2} - b^{2}$

Step 4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = a - 1$ and $b = b$ .

$(a - 1 + b) (a - 1 - b)$