Algebra Examples

$x^{2} - 10 x - 81 - y^{2} - 18 y + 25$

Step 1

Regroup terms.

$x^{2} - 10 x + 25 - 81 - y^{2} - 18 y$

Step 2

Factor using the perfect square rule.

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Step 2.1

Rewrite $25$ as $5^{2}$ .

$x^{2} - 10 x + 5^{2} - 81 - y^{2} - 18 y$

Step 2.2

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

$10 x = 2 \cdot x \cdot 5$

Step 2.3

Rewrite the polynomial.

$x^{2} - 2 \cdot x \cdot 5 + 5^{2} - 81 - y^{2} - 18 y$

Step 2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 5$ .

${(x - 5)}^{2} - 81 - y^{2} - 18 y$

Step 3

Factor by grouping.

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Step 3.1

Reorder terms.

${(x - 5)}^{2} - y^{2} - 18 y - 81$

Step 3.2

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 $a \cdot c = - 1 \cdot - 81 = 81$ and whose sum is $b = - 18$ .

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Step 3.2.1

Factor $- 18$ out of $- 18 y$ .

${(x - 5)}^{2} - y^{2} - 18 (y) - 81$

Step 3.2.2

Rewrite $- 18$ as $- 9$ plus $- 9$

${(x - 5)}^{2} - y^{2} + (- 9 - 9) y - 81$

Step 3.2.3

Apply the distributive property.

${(x - 5)}^{2} - y^{2} - 9 y - 9 y - 81$

Step 3.3

Factor out the greatest common factor from each group.

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Step 3.3.1

Group the first two terms and the last two terms.

${(x - 5)}^{2} + (- y^{2} - 9 y) - 9 y - 81$

Step 3.3.2

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

${(x - 5)}^{2} + y (- y - 9) + 9 (- y - 9)$

Step 3.4

Factor the polynomial by factoring out the greatest common factor, $- y - 9$ .

${(x - 5)}^{2} + (- y - 9) (y + 9)$

Step 4

Rewrite $(y + 9) (y + 9)$ as ${(y + 9)}^{2}$ .

${(x - 5)}^{2} - {(y + 9)}^{2}$

Step 5

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x - 5$ and $b = y + 9$ .

$(x - 5 + y + 9) (x - 5 - (y + 9))$

Step 6

Simplify.

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Step 6.1

Add $- 5$ and $9$ .

$(x + y + 4) (x - 5 - (y + 9))$

Step 6.2

Apply the distributive property.

$(x + y + 4) (x - 5 - y - 1 \cdot 9)$

Step 6.3

Multiply $- 1$ by $9$ .

$(x + y + 4) (x - 5 - y - 9)$

Step 6.4

Subtract $9$ from $- 5$ .

$(x + y + 4) (x - y - 14)$