Basic Math Examples

$20 m n + p^{2} - 4 m^{2} - 25 n^{2}$

Step 1

Regroup terms.

$p^{2} + 20 m n - 4 m^{2} - 25 n^{2}$

Step 2

Add parentheses.

$p^{2} + 20 (m n) - 4 m^{2} - 25 n^{2}$

Step 3

Let $u = m n$ . Substitute $u$ for all occurrences of $m n$ .

$p^{2} + 20 u - 4 m^{2} - 25 n^{2}$

Step 4

Factor $- 1$ out of $20 u - 4 m^{2} - 25 n^{2}$ .

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Step 4.1

Move $20 u$ .

$p^{2} - 4 m^{2} - 25 n^{2} + 20 u$

Step 4.2

Factor $- 1$ out of $- 4 m^{2}$ .

$p^{2} - (4 m^{2}) - 25 n^{2} + 20 u$

Step 4.3

Factor $- 1$ out of $- 25 n^{2}$ .

$p^{2} - (4 m^{2}) - (25 n^{2}) + 20 u$

Step 4.4

Factor $- 1$ out of $20 u$ .

$p^{2} - (4 m^{2}) - (25 n^{2}) - (- 20 u)$

Step 4.5

Factor $- 1$ out of $- (4 m^{2}) - (25 n^{2})$ .

$p^{2} - (4 m^{2} + 25 n^{2}) - (- 20 u)$

Step 4.6

Factor $- 1$ out of $- (4 m^{2} + 25 n^{2}) - (- 20 u)$ .

$p^{2} - (4 m^{2} + 25 n^{2} - 20 u)$

Step 5

Replace all occurrences of $u$ with $m n$ .

$p^{2} - (4 m^{2} + 25 n^{2} - 20 (m n))$

Step 6

Remove parentheses.

$p^{2} - (4 m^{2} + 25 n^{2} - 20 m n)$

Step 7

Factor using the perfect square rule.

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Step 7.1

Rearrange terms.

$p^{2} - (4 m^{2} - 20 m n + 25 n^{2})$

Step 7.2

Rewrite $4 m^{2}$ as ${(2 m)}^{2}$ .

$p^{2} - ({(2 m)}^{2} - 20 m n + 25 n^{2})$

Step 7.3

Rewrite $25 n^{2}$ as ${(5 n)}^{2}$ .

$p^{2} - ({(2 m)}^{2} - 20 m n + {(5 n)}^{2})$

Step 7.4

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

$20 m n = 2 \cdot (2 m) \cdot (5 n)$

Step 7.5

Rewrite the polynomial.

$p^{2} - ({(2 m)}^{2} - 2 \cdot (2 m) \cdot (5 n) + {(5 n)}^{2})$

Step 7.6

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

$p^{2} - {(2 m - 5 n)}^{2}$

Step 8

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

$(p + 2 m - 5 n) (p - (2 m - 5 n))$

Step 9

Simplify.

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Step 9.1

Apply the distributive property.

$(p + 2 m - 5 n) (p - (2 m) - (- 5 n))$

Step 9.2

Multiply $2$ by $- 1$ .

$(p + 2 m - 5 n) (p - 2 m - (- 5 n))$

Step 9.3

Multiply $- 5$ by $- 1$ .

$(p + 2 m - 5 n) (p - 2 m + 5 n)$