Algebra Examples

$10 a^{2} - 27 a b + 5 b^{2}$

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 $a \cdot c = 10 \cdot 5 = 50$ and whose sum is $b = - 27$ .

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Step 1.1.1

Reorder terms.

$10 a^{2} + 5 b^{2} - 27 a b$

Step 1.1.2

Reorder $5 b^{2}$ and $- 27 a b$ .

$10 a^{2} - 27 a b + 5 b^{2}$

Step 1.1.3

Factor $- 27$ out of $- 27 a b$ .

$10 a^{2} - 27 (a b) + 5 b^{2}$

Step 1.1.4

Rewrite $- 27$ as $- 2$ plus $- 25$

$10 a^{2} + (- 2 - 25) (a b) + 5 b^{2}$

Step 1.1.5

Apply the distributive property.

$10 a^{2} - 2 (a b) - 25 (a b) + 5 b^{2}$

Step 1.1.6

Remove unnecessary parentheses.

$10 a^{2} - 2 a b - 25 (a b) + 5 b^{2}$

Step 1.1.7

Remove unnecessary parentheses.

$10 a^{2} - 2 a b - 25 a b + 5 b^{2}$

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.

$(10 a^{2} - 2 a b) - 25 a b + 5 b^{2}$

Step 1.2.2

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

$2 a (5 a - 1 b) - 5 b (5 a - b)$

Step 1.3

Factor the polynomial by factoring out the greatest common factor, $5 a - 1 b$ .

$(5 a - 1 b) (2 a - 5 b)$

Step 2

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

$(5 a - b) (2 a - 5 b)$