Precalculus Examples

$n - n \sqrt{5} = 2$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor $n$ out of $n - n \sqrt{5}$ .

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Step 1.1.1

Raise $n$ to the power of $1$ .

$n - n \sqrt{5} = 2$

Step 1.1.2

Factor $n$ out of $n^{1}$ .

$n \cdot 1 - n \sqrt{5} = 2$

Step 1.1.3

Factor $n$ out of $- n \sqrt{5}$ .

$n \cdot 1 + n (- 1 \sqrt{5}) = 2$

Step 1.1.4

Factor $n$ out of $n \cdot 1 + n (- 1 \sqrt{5})$ .

$n (1 - 1 \sqrt{5}) = 2$

Step 1.2

Rewrite $- 1 \sqrt{5}$ as $- \sqrt{5}$ .

$n (1 - \sqrt{5}) = 2$

Step 2

Divide each term in $n (1 - \sqrt{5}) = 2$ by $1 - \sqrt{5}$ and simplify.

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Step 2.1

Divide each term in $n (1 - \sqrt{5}) = 2$ by $1 - \sqrt{5}$ .

$\frac{n (1 - \sqrt{5})}{1 - \sqrt{5}} = \frac{2}{1 - \sqrt{5}}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $1 - \sqrt{5}$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{n (1 - \sqrt{5})}{1 - \sqrt{5}} = \frac{2}{1 - \sqrt{5}}$

Step 2.2.1.2

Divide $n$ by $1$ .

$n = \frac{2}{1 - \sqrt{5}}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Multiply $\frac{2}{1 - \sqrt{5}}$ by $\frac{1 + \sqrt{5}}{1 + \sqrt{5}}$ .

$n = \frac{2}{1 - \sqrt{5}} \cdot \frac{1 + \sqrt{5}}{1 + \sqrt{5}}$

Step 2.3.2

Multiply $\frac{2}{1 - \sqrt{5}}$ by $\frac{1 + \sqrt{5}}{1 + \sqrt{5}}$ .

$n = \frac{2 (1 + \sqrt{5})}{(1 - \sqrt{5}) (1 + \sqrt{5})}$

Step 2.3.3

Expand the denominator using the FOIL method.

$n = \frac{2 (1 + \sqrt{5})}{1 + \sqrt{5} - \sqrt{5} - {\sqrt{5}}^{2}}$

Step 2.3.4

Simplify.

$n = \frac{2 (1 + \sqrt{5})}{- 4}$

Step 2.3.5

Cancel the common factors.

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Step 2.3.5.1

Factor $2$ out of $- 4$ .

$n = \frac{2 (1 + \sqrt{5})}{2 \cdot - 2}$

Step 2.3.5.2

Cancel the common factor.

$n = \frac{2 (1 + \sqrt{5})}{2 \cdot - 2}$

Step 2.3.5.3

Rewrite the expression.

$n = \frac{1 + \sqrt{5}}{- 2}$

Step 2.3.6

Move the negative in front of the fraction.

$n = - \frac{1 + \sqrt{5}}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$n = - \frac{1 + \sqrt{5}}{2}$

Decimal Form:

$n = - 1.61803398 \dots$