Algebra Examples

$x + \frac{1}{x} = 2$

Step 1

Subtract $2$ from both sides of the equation.

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

Step 2

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

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

Step 3

Combine the numerators over the common denominator.

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

Step 4

Multiply $x$ by $x$ .

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

Step 5

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 6

Combine $- 2$ and $\frac{x}{x}$ .

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Factor using the perfect square rule.

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Step 8.1

Rearrange terms.

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

Step 8.2

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

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

Step 8.3

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

$2 x = 2 \cdot x \cdot 1$

Step 8.4

Rewrite the polynomial.

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

Step 8.5

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

$\frac{{(x - 1)}^{2}}{x} = 0$

Step 9

Set the numerator equal to zero.

${(x - 1)}^{2} = 0$

Step 10

Solve the equation for $x$ .

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Step 10.1

Set the $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 10.2

Add $1$ to both sides of the equation.

$x = 1$