Precalculus Examples

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

Step 1

Set the denominator in $\frac{m^{2} + m - m n - n}{m^{2} + m + m n + n}$ equal to $0$ to find where the expression is undefined.

$m^{2} + m + m n + n = 0$

Step 2

Solve for $m$ .

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Step 2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.2

Substitute the values $a = 1$ , $b = 1 + n$ , and $c = n$ into the quadratic formula and solve for $m$ .

$\frac{- (1 + n) \pm \sqrt{{(1 + n)}^{2} - 4 \cdot (1 \cdot n)}}{2 \cdot 1}$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the numerator.

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Step 2.3.1.1

Apply the distributive property.

$m = \frac{- 1 \cdot 1 - n \pm \sqrt{{(1 + n)}^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.3.1.2

Multiply $- 1$ by $1$ .

$m = \frac{- 1 - n \pm \sqrt{{(1 + n)}^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.3.1.3

Rewrite ${(1 + n)}^{2}$ as $(1 + n) (1 + n)$ .

$m = \frac{- 1 - n \pm \sqrt{(1 + n) (1 + n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.3.1.4

Expand $(1 + n) (1 + n)$ using the FOIL Method.

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Step 2.3.1.4.1

Apply the distributive property.

$m = \frac{- 1 - n \pm \sqrt{1 (1 + n) + n (1 + n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.3.1.4.2

Apply the distributive property.

$m = \frac{- 1 - n \pm \sqrt{1 \cdot 1 + 1 n + n (1 + n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.3.1.4.3

Apply the distributive property.

$m = \frac{- 1 - n \pm \sqrt{1 \cdot 1 + 1 n + n \cdot 1 + n \cdot n - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.3.1.5

Simplify and combine like terms.

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Step 2.3.1.5.1

Simplify each term.

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Step 2.3.1.5.1.1

Multiply $1$ by $1$ .

$m = \frac{- 1 - n \pm \sqrt{1 + 1 n + n \cdot 1 + n \cdot n - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.3.1.5.1.2

Multiply $n$ by $1$ .

$m = \frac{- 1 - n \pm \sqrt{1 + n + n \cdot 1 + n \cdot n - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.3.1.5.1.3

Multiply $n$ by $1$ .

$m = \frac{- 1 - n \pm \sqrt{1 + n + n + n \cdot n - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.3.1.5.1.4

Multiply $n$ by $n$ .

$m = \frac{- 1 - n \pm \sqrt{1 + n + n + n^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.3.1.5.2

Add $n$ and $n$ .

$m = \frac{- 1 - n \pm \sqrt{1 + 2 n + n^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.3.1.6

Multiply $- 4$ by $1$ .

$m = \frac{- 1 - n \pm \sqrt{1 + 2 n + n^{2} - 4 \cdot n}}{2 \cdot 1}$

Step 2.3.1.7

Subtract $4 n$ from $2 n$ .

$m = \frac{- 1 - n \pm \sqrt{1 + n^{2} - 2 n}}{2 \cdot 1}$

Step 2.3.1.8

Reorder terms.

$m = \frac{- 1 - n \pm \sqrt{n^{2} - 2 n + 1}}{2 \cdot 1}$

Step 2.3.1.9

Factor using the perfect square rule.

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Step 2.3.1.9.1

Rewrite $1$ as $1^{2}$ .

$m = \frac{- 1 - n \pm \sqrt{n^{2} - 2 n + 1^{2}}}{2 \cdot 1}$

Step 2.3.1.9.2

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

$2 n = 2 \cdot n \cdot 1$

Step 2.3.1.9.3

Rewrite the polynomial.

$m = \frac{- 1 - n \pm \sqrt{n^{2} - 2 \cdot n \cdot 1 + 1^{2}}}{2 \cdot 1}$

Step 2.3.1.9.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = n$ and $b = 1$ .

$m = \frac{- 1 - n \pm \sqrt{{(n - 1)}^{2}}}{2 \cdot 1}$

Step 2.3.1.10

Pull terms out from under the radical, assuming positive real numbers.

$m = \frac{- 1 - n \pm (n - 1)}{2 \cdot 1}$

Step 2.3.2

Multiply $2$ by $1$ .

$m = \frac{- 1 - n \pm (n - 1)}{2}$

Step 2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

Apply the distributive property.

$m = \frac{- 1 \cdot 1 - n \pm \sqrt{{(1 + n)}^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.4.1.2

Multiply $- 1$ by $1$ .

$m = \frac{- 1 - n \pm \sqrt{{(1 + n)}^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.4.1.3

Rewrite ${(1 + n)}^{2}$ as $(1 + n) (1 + n)$ .

$m = \frac{- 1 - n \pm \sqrt{(1 + n) (1 + n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.4.1.4

Expand $(1 + n) (1 + n)$ using the FOIL Method.

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Step 2.4.1.4.1

Apply the distributive property.

$m = \frac{- 1 - n \pm \sqrt{1 (1 + n) + n (1 + n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.4.1.4.2

Apply the distributive property.

$m = \frac{- 1 - n \pm \sqrt{1 \cdot 1 + 1 n + n (1 + n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.4.1.4.3

Apply the distributive property.

$m = \frac{- 1 - n \pm \sqrt{1 \cdot 1 + 1 n + n \cdot 1 + n \cdot n - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.4.1.5

Simplify and combine like terms.

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Step 2.4.1.5.1

Simplify each term.

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Step 2.4.1.5.1.1

Multiply $1$ by $1$ .

$m = \frac{- 1 - n \pm \sqrt{1 + 1 n + n \cdot 1 + n \cdot n - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.4.1.5.1.2

Multiply $n$ by $1$ .

$m = \frac{- 1 - n \pm \sqrt{1 + n + n \cdot 1 + n \cdot n - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.4.1.5.1.3

Multiply $n$ by $1$ .

$m = \frac{- 1 - n \pm \sqrt{1 + n + n + n \cdot n - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.4.1.5.1.4

Multiply $n$ by $n$ .

$m = \frac{- 1 - n \pm \sqrt{1 + n + n + n^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.4.1.5.2

Add $n$ and $n$ .

$m = \frac{- 1 - n \pm \sqrt{1 + 2 n + n^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.4.1.6

Multiply $- 4$ by $1$ .

$m = \frac{- 1 - n \pm \sqrt{1 + 2 n + n^{2} - 4 \cdot n}}{2 \cdot 1}$

Step 2.4.1.7

Subtract $4 n$ from $2 n$ .

$m = \frac{- 1 - n \pm \sqrt{1 + n^{2} - 2 n}}{2 \cdot 1}$

Step 2.4.1.8

Reorder terms.

$m = \frac{- 1 - n \pm \sqrt{n^{2} - 2 n + 1}}{2 \cdot 1}$

Step 2.4.1.9

Factor using the perfect square rule.

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Step 2.4.1.9.1

Rewrite $1$ as $1^{2}$ .

$m = \frac{- 1 - n \pm \sqrt{n^{2} - 2 n + 1^{2}}}{2 \cdot 1}$

Step 2.4.1.9.2

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

$2 n = 2 \cdot n \cdot 1$

Step 2.4.1.9.3

Rewrite the polynomial.

$m = \frac{- 1 - n \pm \sqrt{n^{2} - 2 \cdot n \cdot 1 + 1^{2}}}{2 \cdot 1}$

Step 2.4.1.9.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = n$ and $b = 1$ .

$m = \frac{- 1 - n \pm \sqrt{{(n - 1)}^{2}}}{2 \cdot 1}$

Step 2.4.1.10

Pull terms out from under the radical, assuming positive real numbers.

$m = \frac{- 1 - n \pm (n - 1)}{2 \cdot 1}$

Step 2.4.2

Multiply $2$ by $1$ .

$m = \frac{- 1 - n \pm (n - 1)}{2}$

Step 2.4.3

Change the $\pm$ to $+$ .

$m = \frac{- 1 - n + n - 1}{2}$

Step 2.4.4

Simplify the numerator.

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Step 2.4.4.1

Subtract $1$ from $- 1$ .

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

Step 2.4.4.2

Add $- n$ and $n$ .

$m = \frac{0 - 2}{2}$

Step 2.4.4.3

Subtract $2$ from $0$ .

$m = \frac{- 2}{2}$

Step 2.4.5

Divide $- 2$ by $2$ .

$m = - 1$

Step 2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Apply the distributive property.

$m = \frac{- 1 \cdot 1 - n \pm \sqrt{{(1 + n)}^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.5.1.2

Multiply $- 1$ by $1$ .

$m = \frac{- 1 - n \pm \sqrt{{(1 + n)}^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.5.1.3

Rewrite ${(1 + n)}^{2}$ as $(1 + n) (1 + n)$ .

$m = \frac{- 1 - n \pm \sqrt{(1 + n) (1 + n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.5.1.4

Expand $(1 + n) (1 + n)$ using the FOIL Method.

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Step 2.5.1.4.1

Apply the distributive property.

$m = \frac{- 1 - n \pm \sqrt{1 (1 + n) + n (1 + n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.5.1.4.2

Apply the distributive property.

$m = \frac{- 1 - n \pm \sqrt{1 \cdot 1 + 1 n + n (1 + n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.5.1.4.3

Apply the distributive property.

$m = \frac{- 1 - n \pm \sqrt{1 \cdot 1 + 1 n + n \cdot 1 + n \cdot n - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.5.1.5

Simplify and combine like terms.

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Step 2.5.1.5.1

Simplify each term.

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Step 2.5.1.5.1.1

Multiply $1$ by $1$ .

$m = \frac{- 1 - n \pm \sqrt{1 + 1 n + n \cdot 1 + n \cdot n - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.5.1.5.1.2

Multiply $n$ by $1$ .

$m = \frac{- 1 - n \pm \sqrt{1 + n + n \cdot 1 + n \cdot n - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.5.1.5.1.3

Multiply $n$ by $1$ .

$m = \frac{- 1 - n \pm \sqrt{1 + n + n + n \cdot n - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.5.1.5.1.4

Multiply $n$ by $n$ .

$m = \frac{- 1 - n \pm \sqrt{1 + n + n + n^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.5.1.5.2

Add $n$ and $n$ .

$m = \frac{- 1 - n \pm \sqrt{1 + 2 n + n^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 2.5.1.6

Multiply $- 4$ by $1$ .

$m = \frac{- 1 - n \pm \sqrt{1 + 2 n + n^{2} - 4 \cdot n}}{2 \cdot 1}$

Step 2.5.1.7

Subtract $4 n$ from $2 n$ .

$m = \frac{- 1 - n \pm \sqrt{1 + n^{2} - 2 n}}{2 \cdot 1}$

Step 2.5.1.8

Reorder terms.

$m = \frac{- 1 - n \pm \sqrt{n^{2} - 2 n + 1}}{2 \cdot 1}$

Step 2.5.1.9

Factor using the perfect square rule.

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Step 2.5.1.9.1

Rewrite $1$ as $1^{2}$ .

$m = \frac{- 1 - n \pm \sqrt{n^{2} - 2 n + 1^{2}}}{2 \cdot 1}$

Step 2.5.1.9.2

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

$2 n = 2 \cdot n \cdot 1$

Step 2.5.1.9.3

Rewrite the polynomial.

$m = \frac{- 1 - n \pm \sqrt{n^{2} - 2 \cdot n \cdot 1 + 1^{2}}}{2 \cdot 1}$

Step 2.5.1.9.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = n$ and $b = 1$ .

$m = \frac{- 1 - n \pm \sqrt{{(n - 1)}^{2}}}{2 \cdot 1}$

Step 2.5.1.10

Pull terms out from under the radical, assuming positive real numbers.

$m = \frac{- 1 - n \pm (n - 1)}{2 \cdot 1}$

Step 2.5.2

Multiply $2$ by $1$ .

$m = \frac{- 1 - n \pm (n - 1)}{2}$

Step 2.5.3

Change the $\pm$ to $-$ .

$m = \frac{- 1 - n - (n - 1)}{2}$

Step 2.5.4

Simplify the numerator.

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Step 2.5.4.1

Apply the distributive property.

$m = \frac{- 1 - n - n + 1}{2}$

Step 2.5.4.2

Multiply $- 1$ by $- 1$ .

$m = \frac{- 1 - n - n + 1}{2}$

Step 2.5.4.3

Add $- 1$ and $1$ .

$m = \frac{- n - n + 0}{2}$

Step 2.5.4.4

Add $- n - n$ and $0$ .

$m = \frac{- n - n}{2}$

Step 2.5.4.5

Subtract $n$ from $- n$ .

$m = \frac{- 2 n}{2}$

Step 2.5.5

Cancel the common factor of $- 2$ and $2$ .

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Step 2.5.5.1

Factor $2$ out of $- 2 n$ .

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

Step 2.5.5.2

Cancel the common factors.

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Step 2.5.5.2.1

Factor $2$ out of $2$ .

$m = \frac{2 (- n)}{2 (1)}$

Step 2.5.5.2.2

Cancel the common factor.

$m = \frac{2 (- n)}{2 \cdot 1}$

Step 2.5.5.2.3

Rewrite the expression.

$m = \frac{- n}{1}$

Step 2.5.5.2.4

Divide $- n$ by $1$ .

$m = - n$

Step 2.6

The final answer is the combination of both solutions.

$m = - 1, - n$

Step 3

Set the denominator in $\frac{m^{2} - m - m n + n}{m^{2} - m + m n - n}$ equal to $0$ to find where the expression is undefined.

$m^{2} - m + m n - n = 0$

Step 4

Solve for $m$ .

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Step 4.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.2

Substitute the values $a = 1$ , $b = - 1 + n$ , and $c = - n$ into the quadratic formula and solve for $m$ .

$\frac{- (- 1 + n) \pm \sqrt{{(- 1 + n)}^{2} - 4 \cdot (1 \cdot (- n))}}{2 \cdot 1}$

Step 4.3

Simplify.

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Step 4.3.1

Simplify the numerator.

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Step 4.3.1.1

Apply the distributive property.

$m = \frac{1 - n \pm \sqrt{{(- 1 + n)}^{2} - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.3.1.2

Multiply $- 1$ by $- 1$ .

$m = \frac{1 - n \pm \sqrt{{(- 1 + n)}^{2} - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.3.1.3

Rewrite ${(- 1 + n)}^{2}$ as $(- 1 + n) (- 1 + n)$ .

$m = \frac{1 - n \pm \sqrt{(- 1 + n) (- 1 + n) - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.3.1.4

Expand $(- 1 + n) (- 1 + n)$ using the FOIL Method.

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Step 4.3.1.4.1

Apply the distributive property.

$m = \frac{1 - n \pm \sqrt{- 1 (- 1 + n) + n (- 1 + n) - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.3.1.4.2

Apply the distributive property.

$m = \frac{1 - n \pm \sqrt{- 1 \cdot - 1 - 1 n + n (- 1 + n) - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.3.1.4.3

Apply the distributive property.

$m = \frac{1 - n \pm \sqrt{- 1 \cdot - 1 - 1 n + n \cdot - 1 + n \cdot n - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.3.1.5

Simplify and combine like terms.

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Step 4.3.1.5.1

Simplify each term.

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Step 4.3.1.5.1.1

Multiply $- 1$ by $- 1$ .

$m = \frac{1 - n \pm \sqrt{1 - 1 n + n \cdot - 1 + n \cdot n - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.3.1.5.1.2

Rewrite $- 1 n$ as $- n$ .

$m = \frac{1 - n \pm \sqrt{1 - n + n \cdot - 1 + n \cdot n - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.3.1.5.1.3

Move $- 1$ to the left of $n$ .

$m = \frac{1 - n \pm \sqrt{1 - n - 1 \cdot n + n \cdot n - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.3.1.5.1.4

Rewrite $- 1 n$ as $- n$ .

$m = \frac{1 - n \pm \sqrt{1 - n - n + n \cdot n - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.3.1.5.1.5

Multiply $n$ by $n$ .

$m = \frac{1 - n \pm \sqrt{1 - n - n + n^{2} - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.3.1.5.2

Subtract $n$ from $- n$ .

$m = \frac{1 - n \pm \sqrt{1 - 2 n + n^{2} - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.3.1.6

Multiply $- 4 \cdot 1 \cdot - 1$ .

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Step 4.3.1.6.1

Multiply $- 4$ by $1$ .

$m = \frac{1 - n \pm \sqrt{1 - 2 n + n^{2} - 4 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.3.1.6.2

Multiply $- 4$ by $- 1$ .

$m = \frac{1 - n \pm \sqrt{1 - 2 n + n^{2} + 4 n}}{2 \cdot 1}$

Step 4.3.1.7

Add $- 2 n$ and $4 n$ .

$m = \frac{1 - n \pm \sqrt{1 + n^{2} + 2 n}}{2 \cdot 1}$

Step 4.3.1.8

Reorder terms.

$m = \frac{1 - n \pm \sqrt{n^{2} + 2 n + 1}}{2 \cdot 1}$

Step 4.3.1.9

Factor using the perfect square rule.

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Step 4.3.1.9.1

Rewrite $1$ as $1^{2}$ .

$m = \frac{1 - n \pm \sqrt{n^{2} + 2 n + 1^{2}}}{2 \cdot 1}$

Step 4.3.1.9.2

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

$2 n = 2 \cdot n \cdot 1$

Step 4.3.1.9.3

Rewrite the polynomial.

$m = \frac{1 - n \pm \sqrt{n^{2} + 2 \cdot n \cdot 1 + 1^{2}}}{2 \cdot 1}$

Step 4.3.1.9.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = n$ and $b = 1$ .

$m = \frac{1 - n \pm \sqrt{{(n + 1)}^{2}}}{2 \cdot 1}$

Step 4.3.1.10

Pull terms out from under the radical, assuming positive real numbers.

$m = \frac{1 - n \pm (n + 1)}{2 \cdot 1}$

Step 4.3.2

Multiply $2$ by $1$ .

$m = \frac{1 - n \pm (n + 1)}{2}$

Step 4.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 4.4.1

Simplify the numerator.

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Step 4.4.1.1

Apply the distributive property.

$m = \frac{1 - n \pm \sqrt{{(- 1 + n)}^{2} - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.4.1.2

Multiply $- 1$ by $- 1$ .

$m = \frac{1 - n \pm \sqrt{{(- 1 + n)}^{2} - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.4.1.3

Rewrite ${(- 1 + n)}^{2}$ as $(- 1 + n) (- 1 + n)$ .

$m = \frac{1 - n \pm \sqrt{(- 1 + n) (- 1 + n) - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.4.1.4

Expand $(- 1 + n) (- 1 + n)$ using the FOIL Method.

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Step 4.4.1.4.1

Apply the distributive property.

$m = \frac{1 - n \pm \sqrt{- 1 (- 1 + n) + n (- 1 + n) - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.4.1.4.2

Apply the distributive property.

$m = \frac{1 - n \pm \sqrt{- 1 \cdot - 1 - 1 n + n (- 1 + n) - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.4.1.4.3

Apply the distributive property.

$m = \frac{1 - n \pm \sqrt{- 1 \cdot - 1 - 1 n + n \cdot - 1 + n \cdot n - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.4.1.5

Simplify and combine like terms.

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Step 4.4.1.5.1

Simplify each term.

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Step 4.4.1.5.1.1

Multiply $- 1$ by $- 1$ .

$m = \frac{1 - n \pm \sqrt{1 - 1 n + n \cdot - 1 + n \cdot n - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.4.1.5.1.2

Rewrite $- 1 n$ as $- n$ .

$m = \frac{1 - n \pm \sqrt{1 - n + n \cdot - 1 + n \cdot n - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.4.1.5.1.3

Move $- 1$ to the left of $n$ .

$m = \frac{1 - n \pm \sqrt{1 - n - 1 \cdot n + n \cdot n - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.4.1.5.1.4

Rewrite $- 1 n$ as $- n$ .

$m = \frac{1 - n \pm \sqrt{1 - n - n + n \cdot n - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.4.1.5.1.5

Multiply $n$ by $n$ .

$m = \frac{1 - n \pm \sqrt{1 - n - n + n^{2} - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.4.1.5.2

Subtract $n$ from $- n$ .

$m = \frac{1 - n \pm \sqrt{1 - 2 n + n^{2} - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.4.1.6

Multiply $- 4 \cdot 1 \cdot - 1$ .

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Step 4.4.1.6.1

Multiply $- 4$ by $1$ .

$m = \frac{1 - n \pm \sqrt{1 - 2 n + n^{2} - 4 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.4.1.6.2

Multiply $- 4$ by $- 1$ .

$m = \frac{1 - n \pm \sqrt{1 - 2 n + n^{2} + 4 n}}{2 \cdot 1}$

Step 4.4.1.7

Add $- 2 n$ and $4 n$ .

$m = \frac{1 - n \pm \sqrt{1 + n^{2} + 2 n}}{2 \cdot 1}$

Step 4.4.1.8

Reorder terms.

$m = \frac{1 - n \pm \sqrt{n^{2} + 2 n + 1}}{2 \cdot 1}$

Step 4.4.1.9

Factor using the perfect square rule.

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Step 4.4.1.9.1

Rewrite $1$ as $1^{2}$ .

$m = \frac{1 - n \pm \sqrt{n^{2} + 2 n + 1^{2}}}{2 \cdot 1}$

Step 4.4.1.9.2

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

$2 n = 2 \cdot n \cdot 1$

Step 4.4.1.9.3

Rewrite the polynomial.

$m = \frac{1 - n \pm \sqrt{n^{2} + 2 \cdot n \cdot 1 + 1^{2}}}{2 \cdot 1}$

Step 4.4.1.9.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = n$ and $b = 1$ .

$m = \frac{1 - n \pm \sqrt{{(n + 1)}^{2}}}{2 \cdot 1}$

Step 4.4.1.10

Pull terms out from under the radical, assuming positive real numbers.

$m = \frac{1 - n \pm (n + 1)}{2 \cdot 1}$

Step 4.4.2

Multiply $2$ by $1$ .

$m = \frac{1 - n \pm (n + 1)}{2}$

Step 4.4.3

Change the $\pm$ to $+$ .

$m = \frac{1 - n + n + 1}{2}$

Step 4.4.4

Simplify the numerator.

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Step 4.4.4.1

Add $1$ and $1$ .

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

Step 4.4.4.2

Add $- n$ and $n$ .

$m = \frac{0 + 2}{2}$

Step 4.4.4.3

Add $0$ and $2$ .

$m = \frac{2}{2}$

Step 4.4.5

Divide $2$ by $2$ .

$m = 1$

Step 4.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 4.5.1

Simplify the numerator.

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Step 4.5.1.1

Apply the distributive property.

$m = \frac{1 - n \pm \sqrt{{(- 1 + n)}^{2} - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.5.1.2

Multiply $- 1$ by $- 1$ .

$m = \frac{1 - n \pm \sqrt{{(- 1 + n)}^{2} - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.5.1.3

Rewrite ${(- 1 + n)}^{2}$ as $(- 1 + n) (- 1 + n)$ .

$m = \frac{1 - n \pm \sqrt{(- 1 + n) (- 1 + n) - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.5.1.4

Expand $(- 1 + n) (- 1 + n)$ using the FOIL Method.

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Step 4.5.1.4.1

Apply the distributive property.

$m = \frac{1 - n \pm \sqrt{- 1 (- 1 + n) + n (- 1 + n) - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.5.1.4.2

Apply the distributive property.

$m = \frac{1 - n \pm \sqrt{- 1 \cdot - 1 - 1 n + n (- 1 + n) - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.5.1.4.3

Apply the distributive property.

$m = \frac{1 - n \pm \sqrt{- 1 \cdot - 1 - 1 n + n \cdot - 1 + n \cdot n - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.5.1.5

Simplify and combine like terms.

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Step 4.5.1.5.1

Simplify each term.

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Step 4.5.1.5.1.1

Multiply $- 1$ by $- 1$ .

$m = \frac{1 - n \pm \sqrt{1 - 1 n + n \cdot - 1 + n \cdot n - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.5.1.5.1.2

Rewrite $- 1 n$ as $- n$ .

$m = \frac{1 - n \pm \sqrt{1 - n + n \cdot - 1 + n \cdot n - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.5.1.5.1.3

Move $- 1$ to the left of $n$ .

$m = \frac{1 - n \pm \sqrt{1 - n - 1 \cdot n + n \cdot n - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.5.1.5.1.4

Rewrite $- 1 n$ as $- n$ .

$m = \frac{1 - n \pm \sqrt{1 - n - n + n \cdot n - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.5.1.5.1.5

Multiply $n$ by $n$ .

$m = \frac{1 - n \pm \sqrt{1 - n - n + n^{2} - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.5.1.5.2

Subtract $n$ from $- n$ .

$m = \frac{1 - n \pm \sqrt{1 - 2 n + n^{2} - 4 \cdot 1 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.5.1.6

Multiply $- 4 \cdot 1 \cdot - 1$ .

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Step 4.5.1.6.1

Multiply $- 4$ by $1$ .

$m = \frac{1 - n \pm \sqrt{1 - 2 n + n^{2} - 4 \cdot (- 1 n)}}{2 \cdot 1}$

Step 4.5.1.6.2

Multiply $- 4$ by $- 1$ .

$m = \frac{1 - n \pm \sqrt{1 - 2 n + n^{2} + 4 n}}{2 \cdot 1}$

Step 4.5.1.7

Add $- 2 n$ and $4 n$ .

$m = \frac{1 - n \pm \sqrt{1 + n^{2} + 2 n}}{2 \cdot 1}$

Step 4.5.1.8

Reorder terms.

$m = \frac{1 - n \pm \sqrt{n^{2} + 2 n + 1}}{2 \cdot 1}$

Step 4.5.1.9

Factor using the perfect square rule.

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Step 4.5.1.9.1

Rewrite $1$ as $1^{2}$ .

$m = \frac{1 - n \pm \sqrt{n^{2} + 2 n + 1^{2}}}{2 \cdot 1}$

Step 4.5.1.9.2

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

$2 n = 2 \cdot n \cdot 1$

Step 4.5.1.9.3

Rewrite the polynomial.

$m = \frac{1 - n \pm \sqrt{n^{2} + 2 \cdot n \cdot 1 + 1^{2}}}{2 \cdot 1}$

Step 4.5.1.9.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = n$ and $b = 1$ .

$m = \frac{1 - n \pm \sqrt{{(n + 1)}^{2}}}{2 \cdot 1}$

Step 4.5.1.10

Pull terms out from under the radical, assuming positive real numbers.

$m = \frac{1 - n \pm (n + 1)}{2 \cdot 1}$

Step 4.5.2

Multiply $2$ by $1$ .

$m = \frac{1 - n \pm (n + 1)}{2}$

Step 4.5.3

Change the $\pm$ to $-$ .

$m = \frac{1 - n - (n + 1)}{2}$

Step 4.5.4

Simplify the numerator.

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Step 4.5.4.1

Apply the distributive property.

$m = \frac{1 - n - n - 1 \cdot 1}{2}$

Step 4.5.4.2

Multiply $- 1$ by $1$ .

$m = \frac{1 - n - n - 1}{2}$

Step 4.5.4.3

Subtract $1$ from $1$ .

$m = \frac{- n - n + 0}{2}$

Step 4.5.4.4

Add $- n - n$ and $0$ .

$m = \frac{- n - n}{2}$

Step 4.5.4.5

Subtract $n$ from $- n$ .

$m = \frac{- 2 n}{2}$

Step 4.5.5

Cancel the common factor of $- 2$ and $2$ .

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Step 4.5.5.1

Factor $2$ out of $- 2 n$ .

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

Step 4.5.5.2

Cancel the common factors.

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Step 4.5.5.2.1

Factor $2$ out of $2$ .

$m = \frac{2 (- n)}{2 (1)}$

Step 4.5.5.2.2

Cancel the common factor.

$m = \frac{2 (- n)}{2 \cdot 1}$

Step 4.5.5.2.3

Rewrite the expression.

$m = \frac{- n}{1}$

Step 4.5.5.2.4

Divide $- n$ by $1$ .

$m = - n$

Step 4.6

The final answer is the combination of both solutions.

$m = 1, - n$

Step 5

Set the denominator in $\frac{m^{2} + m - m n - n}{m^{2} + m + m n + n} \div \frac{m^{2} - m - m n + n}{m^{2} - m + m n - n}$ equal to $0$ to find where the expression is undefined.

$\frac{m^{2} - m - m n + n}{m^{2} - m + m n - n} = 0$

Step 6

Solve for $m$ .

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Step 6.1

Set the numerator equal to zero.

$m^{2} - m - m n + n = 0$

Step 6.2

Solve the equation for $m$ .

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Step 6.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6.2.2

Substitute the values $a = 1$ , $b = - 1 - n$ , and $c = n$ into the quadratic formula and solve for $m$ .

$\frac{- (- 1 - n) \pm \sqrt{{(- 1 - n)}^{2} - 4 \cdot (1 \cdot n)}}{2 \cdot 1}$

Step 6.2.3

Simplify.

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Step 6.2.3.1

Simplify the numerator.

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Step 6.2.3.1.1

Apply the distributive property.

$m = \frac{1 + n \pm \sqrt{{(- 1 - n)}^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.3.1.2

Multiply $- 1$ by $- 1$ .

$m = \frac{1 + n \pm \sqrt{{(- 1 - n)}^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.3.1.3

Multiply $- - n$ .

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Step 6.2.3.1.3.1

Multiply $- 1$ by $- 1$ .

$m = \frac{1 + 1 n \pm \sqrt{{(- 1 - n)}^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.3.1.3.2

Multiply $n$ by $1$ .

$m = \frac{1 + n \pm \sqrt{{(- 1 - n)}^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.3.1.4

Rewrite ${(- 1 - n)}^{2}$ as $(- 1 - n) (- 1 - n)$ .

$m = \frac{1 + n \pm \sqrt{(- 1 - n) (- 1 - n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.3.1.5

Expand $(- 1 - n) (- 1 - n)$ using the FOIL Method.

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Step 6.2.3.1.5.1

Apply the distributive property.

$m = \frac{1 + n \pm \sqrt{- 1 (- 1 - n) - n (- 1 - n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.3.1.5.2

Apply the distributive property.

$m = \frac{1 + n \pm \sqrt{- 1 \cdot - 1 - 1 (- n) - n (- 1 - n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.3.1.5.3

Apply the distributive property.

$m = \frac{1 + n \pm \sqrt{- 1 \cdot - 1 - 1 (- n) - n \cdot - 1 - n (- n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.3.1.6

Simplify and combine like terms.

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Step 6.2.3.1.6.1

Simplify each term.

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Step 6.2.3.1.6.1.1

Multiply $- 1$ by $- 1$ .

$m = \frac{1 + n \pm \sqrt{1 - 1 (- n) - n \cdot - 1 - n (- n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.3.1.6.1.2

Multiply $- 1 (- n)$ .

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Step 6.2.3.1.6.1.2.1

Multiply $- 1$ by $- 1$ .

$m = \frac{1 + n \pm \sqrt{1 + 1 n - n \cdot - 1 - n (- n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.3.1.6.1.2.2

Multiply $n$ by $1$ .

$m = \frac{1 + n \pm \sqrt{1 + n - n \cdot - 1 - n (- n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.3.1.6.1.3

Multiply $- n \cdot - 1$ .

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Step 6.2.3.1.6.1.3.1

Multiply $- 1$ by $- 1$ .

$m = \frac{1 + n \pm \sqrt{1 + n + 1 n - n (- n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.3.1.6.1.3.2

Multiply $n$ by $1$ .

$m = \frac{1 + n \pm \sqrt{1 + n + n - n (- n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.3.1.6.1.4

Rewrite using the commutative property of multiplication.

$m = \frac{1 + n \pm \sqrt{1 + n + n - 1 \cdot (- 1 n \cdot n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.3.1.6.1.5

Multiply $n$ by $n$ by adding the exponents.

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Step 6.2.3.1.6.1.5.1

Move $n$ .

$m = \frac{1 + n \pm \sqrt{1 + n + n - 1 \cdot (- 1 (n \cdot n)) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.3.1.6.1.5.2

Multiply $n$ by $n$ .

$m = \frac{1 + n \pm \sqrt{1 + n + n - 1 \cdot (- 1 n^{2}) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.3.1.6.1.6

Multiply $- 1$ by $- 1$ .

$m = \frac{1 + n \pm \sqrt{1 + n + n + 1 n^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.3.1.6.1.7

Multiply $n^{2}$ by $1$ .

$m = \frac{1 + n \pm \sqrt{1 + n + n + n^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.3.1.6.2

Add $n$ and $n$ .

$m = \frac{1 + n \pm \sqrt{1 + 2 n + n^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.3.1.7

Multiply $- 4$ by $1$ .

$m = \frac{1 + n \pm \sqrt{1 + 2 n + n^{2} - 4 \cdot n}}{2 \cdot 1}$

Step 6.2.3.1.8

Subtract $4 n$ from $2 n$ .

$m = \frac{1 + n \pm \sqrt{1 + n^{2} - 2 n}}{2 \cdot 1}$

Step 6.2.3.1.9

Reorder terms.

$m = \frac{1 + n \pm \sqrt{n^{2} - 2 n + 1}}{2 \cdot 1}$

Step 6.2.3.1.10

Factor using the perfect square rule.

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Step 6.2.3.1.10.1

Rewrite $1$ as $1^{2}$ .

$m = \frac{1 + n \pm \sqrt{n^{2} - 2 n + 1^{2}}}{2 \cdot 1}$

Step 6.2.3.1.10.2

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

$2 n = 2 \cdot n \cdot 1$

Step 6.2.3.1.10.3

Rewrite the polynomial.

$m = \frac{1 + n \pm \sqrt{n^{2} - 2 \cdot n \cdot 1 + 1^{2}}}{2 \cdot 1}$

Step 6.2.3.1.10.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = n$ and $b = 1$ .

$m = \frac{1 + n \pm \sqrt{{(n - 1)}^{2}}}{2 \cdot 1}$

Step 6.2.3.1.11

Pull terms out from under the radical, assuming positive real numbers.

$m = \frac{1 + n \pm (n - 1)}{2 \cdot 1}$

Step 6.2.3.2

Multiply $2$ by $1$ .

$m = \frac{1 + n \pm (n - 1)}{2}$

Step 6.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 6.2.4.1

Simplify the numerator.

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Step 6.2.4.1.1

Apply the distributive property.

$m = \frac{1 + n \pm \sqrt{{(- 1 - n)}^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.4.1.2

Multiply $- 1$ by $- 1$ .

$m = \frac{1 + n \pm \sqrt{{(- 1 - n)}^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.4.1.3

Multiply $- - n$ .

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Step 6.2.4.1.3.1

Multiply $- 1$ by $- 1$ .

$m = \frac{1 + 1 n \pm \sqrt{{(- 1 - n)}^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.4.1.3.2

Multiply $n$ by $1$ .

$m = \frac{1 + n \pm \sqrt{{(- 1 - n)}^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.4.1.4

Rewrite ${(- 1 - n)}^{2}$ as $(- 1 - n) (- 1 - n)$ .

$m = \frac{1 + n \pm \sqrt{(- 1 - n) (- 1 - n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.4.1.5

Expand $(- 1 - n) (- 1 - n)$ using the FOIL Method.

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Step 6.2.4.1.5.1

Apply the distributive property.

$m = \frac{1 + n \pm \sqrt{- 1 (- 1 - n) - n (- 1 - n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.4.1.5.2

Apply the distributive property.

$m = \frac{1 + n \pm \sqrt{- 1 \cdot - 1 - 1 (- n) - n (- 1 - n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.4.1.5.3

Apply the distributive property.

$m = \frac{1 + n \pm \sqrt{- 1 \cdot - 1 - 1 (- n) - n \cdot - 1 - n (- n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.4.1.6

Simplify and combine like terms.

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Step 6.2.4.1.6.1

Simplify each term.

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Step 6.2.4.1.6.1.1

Multiply $- 1$ by $- 1$ .

$m = \frac{1 + n \pm \sqrt{1 - 1 (- n) - n \cdot - 1 - n (- n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.4.1.6.1.2

Multiply $- 1 (- n)$ .

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Step 6.2.4.1.6.1.2.1

Multiply $- 1$ by $- 1$ .

$m = \frac{1 + n \pm \sqrt{1 + 1 n - n \cdot - 1 - n (- n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.4.1.6.1.2.2

Multiply $n$ by $1$ .

$m = \frac{1 + n \pm \sqrt{1 + n - n \cdot - 1 - n (- n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.4.1.6.1.3

Multiply $- n \cdot - 1$ .

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Step 6.2.4.1.6.1.3.1

Multiply $- 1$ by $- 1$ .

$m = \frac{1 + n \pm \sqrt{1 + n + 1 n - n (- n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.4.1.6.1.3.2

Multiply $n$ by $1$ .

$m = \frac{1 + n \pm \sqrt{1 + n + n - n (- n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.4.1.6.1.4

Rewrite using the commutative property of multiplication.

$m = \frac{1 + n \pm \sqrt{1 + n + n - 1 \cdot (- 1 n \cdot n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.4.1.6.1.5

Multiply $n$ by $n$ by adding the exponents.

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Step 6.2.4.1.6.1.5.1

Move $n$ .

$m = \frac{1 + n \pm \sqrt{1 + n + n - 1 \cdot (- 1 (n \cdot n)) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.4.1.6.1.5.2

Multiply $n$ by $n$ .

$m = \frac{1 + n \pm \sqrt{1 + n + n - 1 \cdot (- 1 n^{2}) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.4.1.6.1.6

Multiply $- 1$ by $- 1$ .

$m = \frac{1 + n \pm \sqrt{1 + n + n + 1 n^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.4.1.6.1.7

Multiply $n^{2}$ by $1$ .

$m = \frac{1 + n \pm \sqrt{1 + n + n + n^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.4.1.6.2

Add $n$ and $n$ .

$m = \frac{1 + n \pm \sqrt{1 + 2 n + n^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.4.1.7

Multiply $- 4$ by $1$ .

$m = \frac{1 + n \pm \sqrt{1 + 2 n + n^{2} - 4 \cdot n}}{2 \cdot 1}$

Step 6.2.4.1.8

Subtract $4 n$ from $2 n$ .

$m = \frac{1 + n \pm \sqrt{1 + n^{2} - 2 n}}{2 \cdot 1}$

Step 6.2.4.1.9

Reorder terms.

$m = \frac{1 + n \pm \sqrt{n^{2} - 2 n + 1}}{2 \cdot 1}$

Step 6.2.4.1.10

Factor using the perfect square rule.

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Step 6.2.4.1.10.1

Rewrite $1$ as $1^{2}$ .

$m = \frac{1 + n \pm \sqrt{n^{2} - 2 n + 1^{2}}}{2 \cdot 1}$

Step 6.2.4.1.10.2

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

$2 n = 2 \cdot n \cdot 1$

Step 6.2.4.1.10.3

Rewrite the polynomial.

$m = \frac{1 + n \pm \sqrt{n^{2} - 2 \cdot n \cdot 1 + 1^{2}}}{2 \cdot 1}$

Step 6.2.4.1.10.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = n$ and $b = 1$ .

$m = \frac{1 + n \pm \sqrt{{(n - 1)}^{2}}}{2 \cdot 1}$

Step 6.2.4.1.11

Pull terms out from under the radical, assuming positive real numbers.

$m = \frac{1 + n \pm (n - 1)}{2 \cdot 1}$

Step 6.2.4.2

Multiply $2$ by $1$ .

$m = \frac{1 + n \pm (n - 1)}{2}$

Step 6.2.4.3

Change the $\pm$ to $+$ .

$m = \frac{1 + n + n - 1}{2}$

Step 6.2.4.4

Simplify the numerator.

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Step 6.2.4.4.1

Subtract $1$ from $1$ .

$m = \frac{n + n + 0}{2}$

Step 6.2.4.4.2

Add $n + n$ and $0$ .

$m = \frac{n + n}{2}$

Step 6.2.4.4.3

Add $n$ and $n$ .

$m = \frac{2 n}{2}$

Step 6.2.4.5

Cancel the common factor of $2$ .

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Step 6.2.4.5.1

Cancel the common factor.

$m = \frac{2 n}{2}$

Step 6.2.4.5.2

Divide $n$ by $1$ .

$m = n$

Step 6.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 6.2.5.1

Simplify the numerator.

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Step 6.2.5.1.1

Apply the distributive property.

$m = \frac{1 + n \pm \sqrt{{(- 1 - n)}^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.5.1.2

Multiply $- 1$ by $- 1$ .

$m = \frac{1 + n \pm \sqrt{{(- 1 - n)}^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.5.1.3

Multiply $- - n$ .

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Step 6.2.5.1.3.1

Multiply $- 1$ by $- 1$ .

$m = \frac{1 + 1 n \pm \sqrt{{(- 1 - n)}^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.5.1.3.2

Multiply $n$ by $1$ .

$m = \frac{1 + n \pm \sqrt{{(- 1 - n)}^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.5.1.4

Rewrite ${(- 1 - n)}^{2}$ as $(- 1 - n) (- 1 - n)$ .

$m = \frac{1 + n \pm \sqrt{(- 1 - n) (- 1 - n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.5.1.5

Expand $(- 1 - n) (- 1 - n)$ using the FOIL Method.

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Step 6.2.5.1.5.1

Apply the distributive property.

$m = \frac{1 + n \pm \sqrt{- 1 (- 1 - n) - n (- 1 - n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.5.1.5.2

Apply the distributive property.

$m = \frac{1 + n \pm \sqrt{- 1 \cdot - 1 - 1 (- n) - n (- 1 - n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.5.1.5.3

Apply the distributive property.

$m = \frac{1 + n \pm \sqrt{- 1 \cdot - 1 - 1 (- n) - n \cdot - 1 - n (- n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.5.1.6

Simplify and combine like terms.

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Step 6.2.5.1.6.1

Simplify each term.

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Step 6.2.5.1.6.1.1

Multiply $- 1$ by $- 1$ .

$m = \frac{1 + n \pm \sqrt{1 - 1 (- n) - n \cdot - 1 - n (- n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.5.1.6.1.2

Multiply $- 1 (- n)$ .

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Step 6.2.5.1.6.1.2.1

Multiply $- 1$ by $- 1$ .

$m = \frac{1 + n \pm \sqrt{1 + 1 n - n \cdot - 1 - n (- n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.5.1.6.1.2.2

Multiply $n$ by $1$ .

$m = \frac{1 + n \pm \sqrt{1 + n - n \cdot - 1 - n (- n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.5.1.6.1.3

Multiply $- n \cdot - 1$ .

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Step 6.2.5.1.6.1.3.1

Multiply $- 1$ by $- 1$ .

$m = \frac{1 + n \pm \sqrt{1 + n + 1 n - n (- n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.5.1.6.1.3.2

Multiply $n$ by $1$ .

$m = \frac{1 + n \pm \sqrt{1 + n + n - n (- n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.5.1.6.1.4

Rewrite using the commutative property of multiplication.

$m = \frac{1 + n \pm \sqrt{1 + n + n - 1 \cdot (- 1 n \cdot n) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.5.1.6.1.5

Multiply $n$ by $n$ by adding the exponents.

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Step 6.2.5.1.6.1.5.1

Move $n$ .

$m = \frac{1 + n \pm \sqrt{1 + n + n - 1 \cdot (- 1 (n \cdot n)) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.5.1.6.1.5.2

Multiply $n$ by $n$ .

$m = \frac{1 + n \pm \sqrt{1 + n + n - 1 \cdot (- 1 n^{2}) - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.5.1.6.1.6

Multiply $- 1$ by $- 1$ .

$m = \frac{1 + n \pm \sqrt{1 + n + n + 1 n^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.5.1.6.1.7

Multiply $n^{2}$ by $1$ .

$m = \frac{1 + n \pm \sqrt{1 + n + n + n^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.5.1.6.2

Add $n$ and $n$ .

$m = \frac{1 + n \pm \sqrt{1 + 2 n + n^{2} - 4 \cdot 1 \cdot n}}{2 \cdot 1}$

Step 6.2.5.1.7

Multiply $- 4$ by $1$ .

$m = \frac{1 + n \pm \sqrt{1 + 2 n + n^{2} - 4 \cdot n}}{2 \cdot 1}$

Step 6.2.5.1.8

Subtract $4 n$ from $2 n$ .

$m = \frac{1 + n \pm \sqrt{1 + n^{2} - 2 n}}{2 \cdot 1}$

Step 6.2.5.1.9

Reorder terms.

$m = \frac{1 + n \pm \sqrt{n^{2} - 2 n + 1}}{2 \cdot 1}$

Step 6.2.5.1.10

Factor using the perfect square rule.

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Step 6.2.5.1.10.1

Rewrite $1$ as $1^{2}$ .

$m = \frac{1 + n \pm \sqrt{n^{2} - 2 n + 1^{2}}}{2 \cdot 1}$

Step 6.2.5.1.10.2

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

$2 n = 2 \cdot n \cdot 1$

Step 6.2.5.1.10.3

Rewrite the polynomial.

$m = \frac{1 + n \pm \sqrt{n^{2} - 2 \cdot n \cdot 1 + 1^{2}}}{2 \cdot 1}$

Step 6.2.5.1.10.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = n$ and $b = 1$ .

$m = \frac{1 + n \pm \sqrt{{(n - 1)}^{2}}}{2 \cdot 1}$

Step 6.2.5.1.11

Pull terms out from under the radical, assuming positive real numbers.

$m = \frac{1 + n \pm (n - 1)}{2 \cdot 1}$

Step 6.2.5.2

Multiply $2$ by $1$ .

$m = \frac{1 + n \pm (n - 1)}{2}$

Step 6.2.5.3

Change the $\pm$ to $-$ .

$m = \frac{1 + n - (n - 1)}{2}$

Step 6.2.5.4

Simplify the numerator.

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Step 6.2.5.4.1

Apply the distributive property.

$m = \frac{1 + n - n + 1}{2}$

Step 6.2.5.4.2

Multiply $- 1$ by $- 1$ .

$m = \frac{1 + n - n + 1}{2}$

Step 6.2.5.4.3

Add $1$ and $1$ .

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

Step 6.2.5.4.4

Subtract $n$ from $n$ .

$m = \frac{0 + 2}{2}$

Step 6.2.5.4.5

Add $0$ and $2$ .

$m = \frac{2}{2}$

Step 6.2.5.5

Divide $2$ by $2$ .

$m = 1$

Step 6.2.6

The final answer is the combination of both solutions.

$m = n, 1$

Step 7

The domain is all values of $m$ that make the expression defined.

Set-Builder Notation:

${m | m \neq 1, - 1, m \neq - n, m \neq n}$ , for any integer $n$