Algebra Examples

$2 a^{2} + 2 b^{2} - 5 a b$

Step 1

Factor by grouping.

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Step 1.1

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 $a \cdot c = 2 \cdot 2 = 4$ and whose sum is $b = - 5$ .

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Step 1.1.1

Reorder $2 b^{2}$ and $- 5 a b$ .

$2 a^{2} - 5 a b + 2 b^{2}$

Step 1.1.2

Factor $- 5$ out of $- 5 a b$ .

$2 a^{2} - 5 (a b) + 2 b^{2}$

Step 1.1.3

Rewrite $- 5$ as $- 1$ plus $- 4$

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

Step 1.1.4

Apply the distributive property.

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

Step 1.1.5

Remove unnecessary parentheses.

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

Step 1.1.6

Remove unnecessary parentheses.

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

Step 1.2

Factor out the greatest common factor from each group.

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Step 1.2.1

Group the first two terms and the last two terms.

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

Step 1.2.2

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

$a (2 a - 1 b) - 2 b (2 a - b)$

Step 1.3

Factor the polynomial by factoring out the greatest common factor, $2 a - 1 b$ .

$(2 a - 1 b) (a - 2 b)$

Step 2

Rewrite $- 1 b$ as $- b$ .

$(2 a - b) (a - 2 b)$