Basic Math Examples

$n^{2} - \frac{m^{2}}{n^{2} - 2 m n + m^{2}}$

Step 1

Factor using the perfect square rule.

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Step 1.1

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

$2 m n = 2 \cdot n \cdot m$

Step 1.2

Rewrite the polynomial.

$n^{2} - \frac{m^{2}}{n^{2} - 2 \cdot n \cdot m + m^{2}}$

Step 1.3

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

$n^{2} - \frac{m^{2}}{{(n - m)}^{2}}$

Step 2

Rewrite $n^{2} - \frac{m^{2}}{{(n - m)}^{2}}$ in a factored form.

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Step 2.1

Rewrite $\frac{m^{2}}{{(n - m)}^{2}}$ as ${(\frac{m}{n - m})}^{2}$ .

$n^{2} - {(\frac{m}{n - m})}^{2}$

Step 2.2

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

$(n + \frac{m}{n - m}) (n - \frac{m}{n - m})$

Step 2.3

Simplify.

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Step 2.3.1

To write $n$ as a fraction with a common denominator, multiply by $\frac{n - m}{n - m}$ .

$(\frac{n (n - m)}{n - m} + \frac{m}{n - m}) (n - \frac{m}{n - m})$

Step 2.3.2

Combine the numerators over the common denominator.

$\frac{n (n - m) + m}{n - m} (n - \frac{m}{n - m})$

Step 2.3.3

Rewrite $\frac{n (n - m) + m}{n - m}$ in a factored form.

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Step 2.3.3.1

Apply the distributive property.

$\frac{n \cdot n + n (- m) + m}{n - m} (n - \frac{m}{n - m})$

Step 2.3.3.2

Multiply $n$ by $n$ .

$\frac{n^{2} + n (- m) + m}{n - m} (n - \frac{m}{n - m})$

Step 2.3.3.3

Rewrite using the commutative property of multiplication.

$\frac{n^{2} - n m + m}{n - m} (n - \frac{m}{n - m})$

Step 2.3.4

To write $n$ as a fraction with a common denominator, multiply by $\frac{n - m}{n - m}$ .

$\frac{n^{2} - n m + m}{n - m} (\frac{n (n - m)}{n - m} - \frac{m}{n - m})$

Step 2.3.5

Combine the numerators over the common denominator.

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

Step 2.3.6

Rewrite $\frac{n (n - m) - m}{n - m}$ in a factored form.

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Step 2.3.6.1

Apply the distributive property.

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

Step 2.3.6.2

Multiply $n$ by $n$ .

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

Step 2.3.6.3

Rewrite using the commutative property of multiplication.

$\frac{n^{2} - n m + m}{n - m} \cdot \frac{n^{2} - n m - m}{n - m}$