Algebra Examples

$3 {(a - 2)}^{2} - (2 - a)$

Step 1

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

$3 ((a - 2) (a - 2)) - (2 - a)$

Step 2

Expand $(a - 2) (a - 2)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$3 (a (a - 2) - 2 (a - 2)) - (2 - a)$

Step 2.2

Apply the distributive property.

$3 (a \cdot a + a \cdot - 2 - 2 (a - 2)) - (2 - a)$

Step 2.3

Apply the distributive property.

$3 (a \cdot a + a \cdot - 2 - 2 a - 2 \cdot - 2) - (2 - a)$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $a$ by $a$ .

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

Step 3.1.2

Move $- 2$ to the left of $a$ .

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

Step 3.1.3

Multiply $- 2$ by $- 2$ .

$3 (a^{2} - 2 a - 2 a + 4) - (2 - a)$

Step 3.2

Subtract $2 a$ from $- 2 a$ .

$3 (a^{2} - 4 a + 4) - (2 - a)$

Step 4

Apply the distributive property.

$3 a^{2} + 3 (- 4 a) + 3 \cdot 4 - (2 - a)$

Step 5

Simplify.

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Step 5.1

Multiply $- 4$ by $3$ .

$3 a^{2} - 12 a + 3 \cdot 4 - (2 - a)$

Step 5.2

Multiply $3$ by $4$ .

$3 a^{2} - 12 a + 12 - (2 - a)$

Step 6

Apply the distributive property.

$3 a^{2} - 12 a + 12 - 1 \cdot 2 - - a$

Step 7

Multiply $- 1$ by $2$ .

$3 a^{2} - 12 a + 12 - 2 - - a$

Step 8

Multiply $- - a$ .

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Step 8.1

Multiply $- 1$ by $- 1$ .

$3 a^{2} - 12 a + 12 - 2 + 1 a$

Step 8.2

Multiply $a$ by $1$ .

$3 a^{2} - 12 a + 12 - 2 + a$

Step 9

Add $- 12 a$ and $a$ .

$3 a^{2} - 11 a + 12 - 2$

Step 10

Subtract $2$ from $12$ .

$3 a^{2} - 11 a + 10$

Step 11

Factor by grouping.

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Step 11.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot 10 = 30$ and whose sum is $b = - 11$ .

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Step 11.1.1

Factor $- 11$ out of $- 11 a$ .

$3 a^{2} - 11 (a) + 10$

Step 11.1.2

Rewrite $- 11$ as $- 5$ plus $- 6$

$3 a^{2} + (- 5 - 6) a + 10$

Step 11.1.3

Apply the distributive property.

$3 a^{2} - 5 a - 6 a + 10$

Step 11.2

Factor out the greatest common factor from each group.

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Step 11.2.1

Group the first two terms and the last two terms.

$(3 a^{2} - 5 a) - 6 a + 10$

Step 11.2.2

Factor out the greatest common factor (GCF) from each group.

$a (3 a - 5) - 2 (3 a - 5)$

Step 11.3

Factor the polynomial by factoring out the greatest common factor, $3 a - 5$ .

$(3 a - 5) (a - 2)$