Algebra Examples

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

Step 1

Apply the distributive property.

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

Step 2

Multiply $- 4$ by $1$ .

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

Step 3

Multiply $- 1$ by $- 4$ .

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

Step 4

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

$4 - 4 + 4 a + (1 - a) (1 - a)$

Step 5

Expand $(1 - a) (1 - a)$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$4 - 4 + 4 a + 1 (1 - a) - a (1 - a)$

Step 5.2

Apply the distributive property.

$4 - 4 + 4 a + 1 \cdot 1 + 1 (- a) - a (1 - a)$

Step 5.3

Apply the distributive property.

$4 - 4 + 4 a + 1 \cdot 1 + 1 (- a) - a \cdot 1 - a (- a)$

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $1$ by $1$ .

$4 - 4 + 4 a + 1 + 1 (- a) - a \cdot 1 - a (- a)$

Step 6.1.2

Multiply $- a$ by $1$ .

$4 - 4 + 4 a + 1 - a - a \cdot 1 - a (- a)$

Step 6.1.3

Multiply $- 1$ by $1$ .

$4 - 4 + 4 a + 1 - a - a - a (- a)$

Step 6.1.4

Rewrite using the commutative property of multiplication.

$4 - 4 + 4 a + 1 - a - a - 1 \cdot - 1 a \cdot a$

Step 6.1.5

Multiply $a$ by $a$ by adding the exponents.

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Step 6.1.5.1

Move $a$ .

$4 - 4 + 4 a + 1 - a - a - 1 \cdot - 1 (a \cdot a)$

Step 6.1.5.2

Multiply $a$ by $a$ .

$4 - 4 + 4 a + 1 - a - a - 1 \cdot - 1 a^{2}$

Step 6.1.6

Multiply $- 1$ by $- 1$ .

$4 - 4 + 4 a + 1 - a - a + 1 a^{2}$

Step 6.1.7

Multiply $a^{2}$ by $1$ .

$4 - 4 + 4 a + 1 - a - a + a^{2}$

Step 6.2

Subtract $a$ from $- a$ .

$4 - 4 + 4 a + 1 - 2 a + a^{2}$

Step 7

Subtract $4$ from $4$ .

$0 + 4 a + 1 - 2 a + a^{2}$

Step 8

Add $0$ and $4 a$ .

$4 a + 1 - 2 a + a^{2}$

Step 9

Subtract $2 a$ from $4 a$ .

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

Step 10

Reorder terms.

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

Step 11

Factor using the perfect square rule.

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Step 11.1

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

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

Step 11.2

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 11.3

Rewrite the polynomial.

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

Step 11.4

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

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