Solve by Factoring 1/(x^2)-4/x+1=0

$\frac{1}{x^{2}} - \frac{4}{x} + 1 = 0$

Step 1

Reorder terms.

$- \frac{4}{x} + \frac{1}{x^{2}} + 1 = 0$

Step 2

To write $- \frac{4}{x}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$- \frac{4}{x} \cdot \frac{x}{x} + \frac{1}{x^{2}} + 1 = 0$

Step 3

Write each expression with a common denominator of $x^{2}$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{4}{x}$ by $\frac{x}{x}$ .

$- \frac{4 x}{x \cdot x} + \frac{1}{x^{2}} + 1 = 0$

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

$- \frac{4 x}{x \cdot x} + \frac{1}{x^{2}} + 1 = 0$

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

$- \frac{4 x}{x \cdot x} + \frac{1}{x^{2}} + 1 = 0$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{4 x}{x^{1 + 1}} + \frac{1}{x^{2}} + 1 = 0$

Add $1$ and $1$ .

$- \frac{4 x}{x^{2}} + \frac{1}{x^{2}} + 1 = 0$

Step 4

Combine the numerators over the common denominator.

$\frac{- 4 x + 1}{x^{2}} + 1 = 0$

Step 5

Write $1$ as a fraction with a common denominator.

$\frac{- 4 x + 1}{x^{2}} + \frac{x^{2}}{x^{2}} = 0$

Step 6

Combine the numerators over the common denominator.

$\frac{- 4 x + 1 + x^{2}}{x^{2}} = 0$

Step 7

Reorder terms.

$\frac{x^{2} - 4 x + 1}{x^{2}} = 0$

Step 8

Set the numerator equal to zero.

$x^{2} - 4 x + 1 = 0$

Step 9

Solve the equation for $x$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = - 4$ , and $c = 1$ into the quadratic formula and solve for $x$ .

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

Simplify.

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

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

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1}}{2 \cdot 1}$

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4}}{2 \cdot 1}$

Subtract $4$ from $16$ .

$x = \frac{4 \pm \sqrt{12}}{2 \cdot 1}$

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$x = \frac{4 \pm \sqrt{4 (3)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{4 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{4 \pm 2 \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 \sqrt{3}}{2}$

Simplify $\frac{4 \pm 2 \sqrt{3}}{2}$ .

$x = 2 \pm \sqrt{3}$

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

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

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

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1}}{2 \cdot 1}$

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4}}{2 \cdot 1}$

Subtract $4$ from $16$ .

$x = \frac{4 \pm \sqrt{12}}{2 \cdot 1}$

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$x = \frac{4 \pm \sqrt{4 (3)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{4 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{4 \pm 2 \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 \sqrt{3}}{2}$

Simplify $\frac{4 \pm 2 \sqrt{3}}{2}$ .

$x = 2 \pm \sqrt{3}$

Change the $\pm$ to $+$ .

$x = 2 + \sqrt{3}$

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

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

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

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1}}{2 \cdot 1}$

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4}}{2 \cdot 1}$

Subtract $4$ from $16$ .

$x = \frac{4 \pm \sqrt{12}}{2 \cdot 1}$

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$x = \frac{4 \pm \sqrt{4 (3)}}{2 \cdot 1}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{4 \pm \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{4 \pm 2 \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 \sqrt{3}}{2}$

Simplify $\frac{4 \pm 2 \sqrt{3}}{2}$ .

$x = 2 \pm \sqrt{3}$

Change the $\pm$ to $-$ .

$x = 2 - \sqrt{3}$

The final answer is the combination of both solutions.

$x = 2 + \sqrt{3}, 2 - \sqrt{3}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = 2 + \sqrt{3}, 2 - \sqrt{3}$

Decimal Form:

$x = 3.73205080 \dots, 0.26794919 \dots$

Solve by Factoring 1/(x^2)-4/x+1=0

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