Algebra Examples

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

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 $1$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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 $x$ .

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Step 9.1

Use the quadratic formula to find the solutions.

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

Step 9.2

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

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

Step 9.3

Simplify.

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Step 9.3.1

Simplify the numerator.

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Step 9.3.1.1

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

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

Step 9.3.1.2

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

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Step 9.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 9.3.1.2.2

Multiply $- 4$ by $1$ .

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

Step 9.3.1.3

Subtract $4$ from $16$ .

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

Step 9.3.1.4

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

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Step 9.3.1.4.1

Factor $4$ out of $12$ .

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

Step 9.3.1.4.2

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

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

Step 9.3.1.5

Pull terms out from under the radical.

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

Step 9.3.2

Multiply $2$ by $1$ .

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

Step 9.3.3

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

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

Step 9.4

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

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Step 9.4.1

Simplify the numerator.

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Step 9.4.1.1

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

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

Step 9.4.1.2

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

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Step 9.4.1.2.1

Multiply $- 4$ by $1$ .

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

Step 9.4.1.2.2

Multiply $- 4$ by $1$ .

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

Step 9.4.1.3

Subtract $4$ from $16$ .

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

Step 9.4.1.4

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

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Step 9.4.1.4.1

Factor $4$ out of $12$ .

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

Step 9.4.1.4.2

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

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

Step 9.4.1.5

Pull terms out from under the radical.

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

Step 9.4.2

Multiply $2$ by $1$ .

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

Step 9.4.3

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

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

Step 9.4.4

Change the $\pm$ to $+$ .

$x = 2 + \sqrt{3}$

Step 9.5

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

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Step 9.5.1

Simplify the numerator.

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Step 9.5.1.1

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

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

Step 9.5.1.2

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

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Step 9.5.1.2.1

Multiply $- 4$ by $1$ .

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

Step 9.5.1.2.2

Multiply $- 4$ by $1$ .

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

Step 9.5.1.3

Subtract $4$ from $16$ .

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

Step 9.5.1.4

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

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Step 9.5.1.4.1

Factor $4$ out of $12$ .

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

Step 9.5.1.4.2

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

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

Step 9.5.1.5

Pull terms out from under the radical.

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

Step 9.5.2

Multiply $2$ by $1$ .

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

Step 9.5.3

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

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

Step 9.5.4

Change the $\pm$ to $-$ .

$x = 2 - \sqrt{3}$

Step 9.6

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:

$x = 3.73205080 \dots, 0.26794919 \dots$