Algebra Examples

$\frac{a^{2}}{4} - a b + b^{2}$

Step 1

Factor $- 1$ out of $\frac{a^{2}}{4} - a b + b^{2}$ .

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Step 1.1

Factor $- 1$ out of $\frac{a^{2}}{4}$ .

$- \frac{- a^{2}}{4} - a b + b^{2}$

Step 1.2

Factor $- 1$ out of $- a b$ .

$- \frac{- a^{2}}{4} - (a b) + b^{2}$

Step 1.3

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

$- \frac{- a^{2}}{4} - (a b) - 1 (- b^{2})$

Step 1.4

Factor $- 1$ out of $- \frac{- a^{2}}{4} - (a b)$ .

$- (\frac{- a^{2}}{4} + a b) - 1 (- b^{2})$

Step 1.5

Factor $- 1$ out of $- (\frac{- a^{2}}{4} + a b) - 1 (- b^{2})$ .

$- (\frac{- a^{2}}{4} + a b - b^{2})$

Step 2

Factor.

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Step 2.1

Factor $\frac{1}{4}$ out of each term.

$- (\frac{1}{4} (- a^{2} + 4 a b - 4 b^{2}))$

Step 2.2

Remove unnecessary parentheses.

$- \frac{1}{4} (- a^{2} + 4 a b - 4 b^{2})$

Step 3

Factor $- 1$ out of $- a^{2} + 4 a b - 4 b^{2}$ .

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Step 3.1

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

$- \frac{1}{4} \cdot (- (a^{2}) + 4 a b - 4 b^{2})$

Step 3.2

Factor $- 1$ out of $4 a b$ .

$- \frac{1}{4} \cdot (- (a^{2}) - (- 4 a b) - 4 b^{2})$

Step 3.3

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

$- \frac{1}{4} \cdot (- (a^{2}) - (- 4 a b) - (4 b^{2}))$

Step 3.4

Factor $- 1$ out of $- (a^{2}) - (- 4 a b)$ .

$- \frac{1}{4} \cdot (- (a^{2} - 4 a b) - (4 b^{2}))$

Step 3.5

Factor $- 1$ out of $- (a^{2} - 4 a b) - (4 b^{2})$ .

$- \frac{1}{4} \cdot (- (a^{2} - 4 a b + 4 b^{2}))$

Step 4

Factor.

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Step 4.1

Factor using the perfect square rule.

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Step 4.1.1

Rewrite $4 b^{2}$ as ${(2 b)}^{2}$ .

$- \frac{1}{4} \cdot (- (a^{2} - 4 a b + {(2 b)}^{2}))$

Step 4.1.2

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

$4 a b = 2 \cdot a \cdot (2 b)$

Step 4.1.3

Rewrite the polynomial.

$- \frac{1}{4} \cdot (- (a^{2} - 2 \cdot a \cdot (2 b) + {(2 b)}^{2}))$

Step 4.1.4

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

$- \frac{1}{4} \cdot (- {(a - 2 b)}^{2})$

Step 4.2

Remove unnecessary parentheses.

$- \frac{1}{4} \cdot - 1 {(a - 2 b)}^{2}$

Step 5

Combine exponents.

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Step 5.1

Factor out negative.

$- (\frac{1}{4} \cdot - 1 {(a - 2 b)}^{2})$

Step 5.2

Combine $\frac{1}{4}$ and $- 1$ .

$- (\frac{- 1}{4} {(a - 2 b)}^{2})$

Step 5.3

Combine $\frac{- 1}{4}$ and ${(a - 2 b)}^{2}$ .

$- \frac{- {(a - 2 b)}^{2}}{4}$