Basic Math Examples

a^{2} - d^{2} + n^{2} - c^{2} - 2 a n - 2 c d

$a^{2} - d^{2} + n^{2} - c^{2} - 2 a n - 2 c d$

Step 1

Regroup terms.

a^{2} + n^{2} - 2 a n - d^{2} - c^{2} - 2 c d

$a^{2} + n^{2} - 2 a n - d^{2} - c^{2} - 2 c d$

Step 2

Factor using the perfect square rule.

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Step 2.1

Rearrange terms.

a^{2} - 2 a n + n^{2} - d^{2} - c^{2} - 2 c d

$a^{2} - 2 a n + n^{2} - d^{2} - c^{2} - 2 c d$

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.

2 a n = 2 \cdot a \cdot n

$2 a n = 2 \cdot a \cdot n$

Step 2.3

Rewrite the polynomial.

a^{2} - 2 \cdot a \cdot n + n^{2} - d^{2} - c^{2} - 2 c d

$a^{2} - 2 \cdot a \cdot n + n^{2} - d^{2} - c^{2} - 2 c d$

Step 2.4

Factor using the perfect square trinomial rule

a^{2} - 2 a b + b^{2} = {(a - b)}^{2}

$a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where

a = a

$a = a$ and

b = n

$b = n$ .

{(a - n)}^{2} - d^{2} - c^{2} - 2 c d

${(a - n)}^{2} - d^{2} - c^{2} - 2 c d$

{(a - n)}^{2} - d^{2} - c^{2} - 2 c d

${(a - n)}^{2} - d^{2} - c^{2} - 2 c d$

Step 3

Factor by grouping.

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Step 3.1

For a polynomial of the form

a x^{2} + b x + c

$a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is

a \cdot c = - 1 \cdot - 1 = 1

$a \cdot c = - 1 \cdot - 1 = 1$ and whose sum is

b = - 2

$b = - 2$ .

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Step 3.1.1

Reorder terms.

{(a - n)}^{2} - c^{2} - d^{2} - 2 c d

${(a - n)}^{2} - c^{2} - d^{2} - 2 c d$

Step 3.1.2

Reorder

- d^{2}

$- d^{2}$ and

- 2 c d

$- 2 c d$ .

{(a - n)}^{2} - c^{2} - 2 c d - d^{2}

${(a - n)}^{2} - c^{2} - 2 c d - d^{2}$

Step 3.1.3

Factor

- 2

$- 2$ out of

- 2 c d

$- 2 c d$ .

{(a - n)}^{2} - c^{2} - 2 (c d) - d^{2}

${(a - n)}^{2} - c^{2} - 2 (c d) - d^{2}$

Step 3.1.4

Rewrite

- 2

$- 2$ as

- 1

$- 1$ plus

- 1

$- 1$

{(a - n)}^{2} - c^{2} + (- 1 - 1) (c d) - d^{2}

${(a - n)}^{2} - c^{2} + (- 1 - 1) (c d) - d^{2}$

Step 3.1.5

Apply the distributive property.

{(a - n)}^{2} - c^{2} - 1 (c d) - 1 (c d) - d^{2}

${(a - n)}^{2} - c^{2} - 1 (c d) - 1 (c d) - d^{2}$

Step 3.1.6

Remove unnecessary parentheses.

{(a - n)}^{2} - c^{2} - 1 c d - 1 (c d) - d^{2}

${(a - n)}^{2} - c^{2} - 1 c d - 1 (c d) - d^{2}$

Step 3.1.7

Remove unnecessary parentheses.

{(a - n)}^{2} - c^{2} - 1 c d - 1 c d - d^{2}

${(a - n)}^{2} - c^{2} - 1 c d - 1 c d - d^{2}$

{(a - n)}^{2} - c^{2} - 1 c d - 1 c d - d^{2}

${(a - n)}^{2} - c^{2} - 1 c d - 1 c d - d^{2}$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

{(a - n)}^{2} + (- c^{2} - 1 c d) - 1 c d - d^{2}

${(a - n)}^{2} + (- c^{2} - 1 c d) - 1 c d - d^{2}$

Step 3.2.2

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

${(a - n)}^{2} + c (- c - 1 d) + d (- 1 c - d)$

Step 3.3

Factor the polynomial by factoring out the greatest common factor,

$- c - 1 d$ .

${(a - n)}^{2} + (- c - 1 d) (c + d)$

Step 4

Rewrite

$- 1 d$ as

$- d$ .

${(a - n)}^{2} + (- c - d) (c + d)$

Step 5

Rewrite

$(c + d) (c + d)$ as

${(c + d)}^{2}$ .

${(a - n)}^{2} - {(c + d)}^{2}$

Step 6

Since both terms are perfect squares, factor using the difference of squares formula,

$a^{2} - b^{2} = (a + b) (a - b)$ where

$a = a - n$ and

$b = c + d$ .

$(a - n + c + d) (a - n - (c + d))$

Step 7

Apply the distributive property.

$(a - n + c + d) (a - n - c - d)$