Solve x=1/x+1/6

$$x = \frac { 1 } { x } + \frac { 1 } { 6 }$$

Solve for x

$$x = \frac{\sqrt{145} + 1}{12} \approx 1.086799548$$

$$x=\frac{1-\sqrt{145}}{12}\approx -0.920132882$$

Steps Using the Quadratic Formula

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and 6 is 6x. Multiply \frac{1}{x} times \frac{6}{6}. Multiply \frac{1}{6} times \frac{x}{x}.

$$x=\frac{6}{6x}+\frac{x}{6x}$$

Since \frac{6}{6x} and \frac{x}{6x} have the same denominator, add them by adding their numerators.

$$x=\frac{6+x}{6x}$$

Subtract \frac{6+x}{6x} from both sides.

$$x-\frac{6+x}{6x}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{6x}{6x}.

$$\frac{x\times 6x}{6x}-\frac{6+x}{6x}=0$$

Since \frac{x\times 6x}{6x} and \frac{6+x}{6x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x\times 6x-\left(6+x\right)}{6x}=0$$

Do the multiplications in x\times 6x-\left(6+x\right).

$$\frac{6x^{2}-6-x}{6x}=0$$

Factor the expressions that are not already factored in \frac{6x^{2}-6-x}{6x}.

$$\frac{6\left(x-\left(-\frac{1}{12}\sqrt{145}+\frac{1}{12}\right)\right)\left(x-\left(\frac{1}{12}\sqrt{145}+\frac{1}{12}\right)\right)}{6x}=0$$

Cancel out 6 in both numerator and denominator.

$$\frac{\left(x-\left(-\frac{1}{12}\sqrt{145}+\frac{1}{12}\right)\right)\left(x-\left(\frac{1}{12}\sqrt{145}+\frac{1}{12}\right)\right)}{x}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$\left(x-\left(-\frac{1}{12}\sqrt{145}+\frac{1}{12}\right)\right)\left(x-\left(\frac{1}{12}\sqrt{145}+\frac{1}{12}\right)\right)=0$$

To find the opposite of -\frac{1}{12}\sqrt{145}+\frac{1}{12}, find the opposite of each term.

$$\left(x-\left(-\frac{1}{12}\sqrt{145}\right)-\frac{1}{12}\right)\left(x-\left(\frac{1}{12}\sqrt{145}+\frac{1}{12}\right)\right)=0$$

The opposite of -\frac{1}{12}\sqrt{145} is \frac{1}{12}\sqrt{145}.

$$\left(x+\frac{1}{12}\sqrt{145}-\frac{1}{12}\right)\left(x-\left(\frac{1}{12}\sqrt{145}+\frac{1}{12}\right)\right)=0$$

To find the opposite of \frac{1}{12}\sqrt{145}+\frac{1}{12}, find the opposite of each term.

$$\left(x+\frac{1}{12}\sqrt{145}-\frac{1}{12}\right)\left(x-\frac{1}{12}\sqrt{145}-\frac{1}{12}\right)=0$$

Apply the distributive property by multiplying each term of x+\frac{1}{12}\sqrt{145}-\frac{1}{12} by each term of x-\frac{1}{12}\sqrt{145}-\frac{1}{12}.

$$x^{2}+x\left(-\frac{1}{12}\right)\sqrt{145}+x\left(-\frac{1}{12}\right)+\frac{1}{12}\sqrt{145}x+\frac{1}{12}\sqrt{145}\left(-\frac{1}{12}\right)\sqrt{145}+\frac{1}{12}\sqrt{145}\left(-\frac{1}{12}\right)-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Multiply \sqrt{145} and \sqrt{145} to get 145.

$$x^{2}+x\left(-\frac{1}{12}\right)\sqrt{145}+x\left(-\frac{1}{12}\right)+\frac{1}{12}\sqrt{145}x+\frac{1}{12}\times 145\left(-\frac{1}{12}\right)+\frac{1}{12}\sqrt{145}\left(-\frac{1}{12}\right)-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Combine x\left(-\frac{1}{12}\right)\sqrt{145} and \frac{1}{12}\sqrt{145}x to get 0.

$$x^{2}+x\left(-\frac{1}{12}\right)+\frac{1}{12}\times 145\left(-\frac{1}{12}\right)+\frac{1}{12}\sqrt{145}\left(-\frac{1}{12}\right)-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Multiply \frac{1}{12} and 145 to get \frac{145}{12}.

$$x^{2}+x\left(-\frac{1}{12}\right)+\frac{145}{12}\left(-\frac{1}{12}\right)+\frac{1}{12}\sqrt{145}\left(-\frac{1}{12}\right)-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Multiply \frac{145}{12} times -\frac{1}{12} by multiplying numerator times numerator and denominator times denominator.

$$x^{2}+x\left(-\frac{1}{12}\right)+\frac{145\left(-1\right)}{12\times 12}+\frac{1}{12}\sqrt{145}\left(-\frac{1}{12}\right)-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Do the multiplications in the fraction \frac{145\left(-1\right)}{12\times 12}.

$$x^{2}+x\left(-\frac{1}{12}\right)+\frac{-145}{144}+\frac{1}{12}\sqrt{145}\left(-\frac{1}{12}\right)-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Fraction \frac{-145}{144} can be rewritten as -\frac{145}{144} by extracting the negative sign.

$$x^{2}+x\left(-\frac{1}{12}\right)-\frac{145}{144}+\frac{1}{12}\sqrt{145}\left(-\frac{1}{12}\right)-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Multiply \frac{1}{12} times -\frac{1}{12} by multiplying numerator times numerator and denominator times denominator.

$$x^{2}+x\left(-\frac{1}{12}\right)-\frac{145}{144}+\frac{1\left(-1\right)}{12\times 12}\sqrt{145}-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Do the multiplications in the fraction \frac{1\left(-1\right)}{12\times 12}.

$$x^{2}+x\left(-\frac{1}{12}\right)-\frac{145}{144}+\frac{-1}{144}\sqrt{145}-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Fraction \frac{-1}{144} can be rewritten as -\frac{1}{144} by extracting the negative sign.

$$x^{2}+x\left(-\frac{1}{12}\right)-\frac{145}{144}-\frac{1}{144}\sqrt{145}-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Combine x\left(-\frac{1}{12}\right) and -\frac{1}{12}x to get -\frac{1}{6}x.

$$x^{2}-\frac{1}{6}x-\frac{145}{144}-\frac{1}{144}\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Multiply -\frac{1}{12} times -\frac{1}{12} by multiplying numerator times numerator and denominator times denominator.

$$x^{2}-\frac{1}{6}x-\frac{145}{144}-\frac{1}{144}\sqrt{145}+\frac{-\left(-1\right)}{12\times 12}\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Do the multiplications in the fraction \frac{-\left(-1\right)}{12\times 12}.

$$x^{2}-\frac{1}{6}x-\frac{145}{144}-\frac{1}{144}\sqrt{145}+\frac{1}{144}\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Combine -\frac{1}{144}\sqrt{145} and \frac{1}{144}\sqrt{145} to get 0.

$$x^{2}-\frac{1}{6}x-\frac{145}{144}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Multiply -\frac{1}{12} times -\frac{1}{12} by multiplying numerator times numerator and denominator times denominator.

$$x^{2}-\frac{1}{6}x-\frac{145}{144}+\frac{-\left(-1\right)}{12\times 12}=0$$

Do the multiplications in the fraction \frac{-\left(-1\right)}{12\times 12}.

$$x^{2}-\frac{1}{6}x-\frac{145}{144}+\frac{1}{144}=0$$

Since -\frac{145}{144} and \frac{1}{144} have the same denominator, add them by adding their numerators.

$$x^{2}-\frac{1}{6}x+\frac{-145+1}{144}=0$$

Add -145 and 1 to get -144.

$$x^{2}-\frac{1}{6}x+\frac{-144}{144}=0$$

Divide -144 by 144 to get -1.

$$x^{2}-\frac{1}{6}x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -\frac{1}{6} for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{1}{6}\right)±\sqrt{\left(-\frac{1}{6}\right)^{2}-4\left(-1\right)}}{2}$$

Square -\frac{1}{6} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{1}{6}\right)±\sqrt{\frac{1}{36}-4\left(-1\right)}}{2}$$

Multiply -4 times -1.

$$x=\frac{-\left(-\frac{1}{6}\right)±\sqrt{\frac{1}{36}+4}}{2}$$

Add \frac{1}{36} to 4.

$$x=\frac{-\left(-\frac{1}{6}\right)±\sqrt{\frac{145}{36}}}{2}$$

Take the square root of \frac{145}{36}.

$$x=\frac{-\left(-\frac{1}{6}\right)±\frac{\sqrt{145}}{6}}{2}$$

The opposite of -\frac{1}{6} is \frac{1}{6}.

$$x=\frac{\frac{1}{6}±\frac{\sqrt{145}}{6}}{2}$$

Now solve the equation x=\frac{\frac{1}{6}±\frac{\sqrt{145}}{6}}{2} when ± is plus. Add \frac{1}{6} to \frac{\sqrt{145}}{6}.

$$x=\frac{\sqrt{145}+1}{2\times 6}$$

Divide \frac{1+\sqrt{145}}{6} by 2.

$$x=\frac{\sqrt{145}+1}{12}$$

Now solve the equation x=\frac{\frac{1}{6}±\frac{\sqrt{145}}{6}}{2} when ± is minus. Subtract \frac{\sqrt{145}}{6} from \frac{1}{6}.

$$x=\frac{1-\sqrt{145}}{2\times 6}$$

Divide \frac{1-\sqrt{145}}{6} by 2.

$$x=\frac{1-\sqrt{145}}{12}$$

The equation is now solved.

$$x=\frac{\sqrt{145}+1}{12} x=\frac{1-\sqrt{145}}{12}$$

Steps for Completing the Square

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and 6 is 6x. Multiply \frac{1}{x} times \frac{6}{6}. Multiply \frac{1}{6} times \frac{x}{x}.

$$x=\frac{6}{6x}+\frac{x}{6x}$$

Since \frac{6}{6x} and \frac{x}{6x} have the same denominator, add them by adding their numerators.

$$x=\frac{6+x}{6x}$$

Subtract \frac{6+x}{6x} from both sides.

$$x-\frac{6+x}{6x}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{6x}{6x}.

$$\frac{x\times 6x}{6x}-\frac{6+x}{6x}=0$$

Since \frac{x\times 6x}{6x} and \frac{6+x}{6x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x\times 6x-\left(6+x\right)}{6x}=0$$

Do the multiplications in x\times 6x-\left(6+x\right).

$$\frac{6x^{2}-6-x}{6x}=0$$

Factor the expressions that are not already factored in \frac{6x^{2}-6-x}{6x}.

$$\frac{6\left(x-\left(-\frac{1}{12}\sqrt{145}+\frac{1}{12}\right)\right)\left(x-\left(\frac{1}{12}\sqrt{145}+\frac{1}{12}\right)\right)}{6x}=0$$

Cancel out 6 in both numerator and denominator.

$$\frac{\left(x-\left(-\frac{1}{12}\sqrt{145}+\frac{1}{12}\right)\right)\left(x-\left(\frac{1}{12}\sqrt{145}+\frac{1}{12}\right)\right)}{x}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$\left(x-\left(-\frac{1}{12}\sqrt{145}+\frac{1}{12}\right)\right)\left(x-\left(\frac{1}{12}\sqrt{145}+\frac{1}{12}\right)\right)=0$$

To find the opposite of -\frac{1}{12}\sqrt{145}+\frac{1}{12}, find the opposite of each term.

$$\left(x-\left(-\frac{1}{12}\sqrt{145}\right)-\frac{1}{12}\right)\left(x-\left(\frac{1}{12}\sqrt{145}+\frac{1}{12}\right)\right)=0$$

The opposite of -\frac{1}{12}\sqrt{145} is \frac{1}{12}\sqrt{145}.

$$\left(x+\frac{1}{12}\sqrt{145}-\frac{1}{12}\right)\left(x-\left(\frac{1}{12}\sqrt{145}+\frac{1}{12}\right)\right)=0$$

To find the opposite of \frac{1}{12}\sqrt{145}+\frac{1}{12}, find the opposite of each term.

$$\left(x+\frac{1}{12}\sqrt{145}-\frac{1}{12}\right)\left(x-\frac{1}{12}\sqrt{145}-\frac{1}{12}\right)=0$$

Apply the distributive property by multiplying each term of x+\frac{1}{12}\sqrt{145}-\frac{1}{12} by each term of x-\frac{1}{12}\sqrt{145}-\frac{1}{12}.

$$x^{2}+x\left(-\frac{1}{12}\right)\sqrt{145}+x\left(-\frac{1}{12}\right)+\frac{1}{12}\sqrt{145}x+\frac{1}{12}\sqrt{145}\left(-\frac{1}{12}\right)\sqrt{145}+\frac{1}{12}\sqrt{145}\left(-\frac{1}{12}\right)-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Multiply \sqrt{145} and \sqrt{145} to get 145.

$$x^{2}+x\left(-\frac{1}{12}\right)\sqrt{145}+x\left(-\frac{1}{12}\right)+\frac{1}{12}\sqrt{145}x+\frac{1}{12}\times 145\left(-\frac{1}{12}\right)+\frac{1}{12}\sqrt{145}\left(-\frac{1}{12}\right)-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Combine x\left(-\frac{1}{12}\right)\sqrt{145} and \frac{1}{12}\sqrt{145}x to get 0.

$$x^{2}+x\left(-\frac{1}{12}\right)+\frac{1}{12}\times 145\left(-\frac{1}{12}\right)+\frac{1}{12}\sqrt{145}\left(-\frac{1}{12}\right)-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Multiply \frac{1}{12} and 145 to get \frac{145}{12}.

$$x^{2}+x\left(-\frac{1}{12}\right)+\frac{145}{12}\left(-\frac{1}{12}\right)+\frac{1}{12}\sqrt{145}\left(-\frac{1}{12}\right)-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Multiply \frac{145}{12} times -\frac{1}{12} by multiplying numerator times numerator and denominator times denominator.

$$x^{2}+x\left(-\frac{1}{12}\right)+\frac{145\left(-1\right)}{12\times 12}+\frac{1}{12}\sqrt{145}\left(-\frac{1}{12}\right)-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Do the multiplications in the fraction \frac{145\left(-1\right)}{12\times 12}.

$$x^{2}+x\left(-\frac{1}{12}\right)+\frac{-145}{144}+\frac{1}{12}\sqrt{145}\left(-\frac{1}{12}\right)-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Fraction \frac{-145}{144} can be rewritten as -\frac{145}{144} by extracting the negative sign.

$$x^{2}+x\left(-\frac{1}{12}\right)-\frac{145}{144}+\frac{1}{12}\sqrt{145}\left(-\frac{1}{12}\right)-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Multiply \frac{1}{12} times -\frac{1}{12} by multiplying numerator times numerator and denominator times denominator.

$$x^{2}+x\left(-\frac{1}{12}\right)-\frac{145}{144}+\frac{1\left(-1\right)}{12\times 12}\sqrt{145}-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Do the multiplications in the fraction \frac{1\left(-1\right)}{12\times 12}.

$$x^{2}+x\left(-\frac{1}{12}\right)-\frac{145}{144}+\frac{-1}{144}\sqrt{145}-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Fraction \frac{-1}{144} can be rewritten as -\frac{1}{144} by extracting the negative sign.

$$x^{2}+x\left(-\frac{1}{12}\right)-\frac{145}{144}-\frac{1}{144}\sqrt{145}-\frac{1}{12}x-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Combine x\left(-\frac{1}{12}\right) and -\frac{1}{12}x to get -\frac{1}{6}x.

$$x^{2}-\frac{1}{6}x-\frac{145}{144}-\frac{1}{144}\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Multiply -\frac{1}{12} times -\frac{1}{12} by multiplying numerator times numerator and denominator times denominator.

$$x^{2}-\frac{1}{6}x-\frac{145}{144}-\frac{1}{144}\sqrt{145}+\frac{-\left(-1\right)}{12\times 12}\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Do the multiplications in the fraction \frac{-\left(-1\right)}{12\times 12}.

$$x^{2}-\frac{1}{6}x-\frac{145}{144}-\frac{1}{144}\sqrt{145}+\frac{1}{144}\sqrt{145}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Combine -\frac{1}{144}\sqrt{145} and \frac{1}{144}\sqrt{145} to get 0.

$$x^{2}-\frac{1}{6}x-\frac{145}{144}-\frac{1}{12}\left(-\frac{1}{12}\right)=0$$

Multiply -\frac{1}{12} times -\frac{1}{12} by multiplying numerator times numerator and denominator times denominator.

$$x^{2}-\frac{1}{6}x-\frac{145}{144}+\frac{-\left(-1\right)}{12\times 12}=0$$

Do the multiplications in the fraction \frac{-\left(-1\right)}{12\times 12}.

$$x^{2}-\frac{1}{6}x-\frac{145}{144}+\frac{1}{144}=0$$

Since -\frac{145}{144} and \frac{1}{144} have the same denominator, add them by adding their numerators.

$$x^{2}-\frac{1}{6}x+\frac{-145+1}{144}=0$$

Add -145 and 1 to get -144.

$$x^{2}-\frac{1}{6}x+\frac{-144}{144}=0$$

Divide -144 by 144 to get -1.

$$x^{2}-\frac{1}{6}x-1=0$$

Add 1 to both sides. Anything plus zero gives itself.

$$x^{2}-\frac{1}{6}x=1$$

Divide -\frac{1}{6}, the coefficient of the x term, by 2 to get -\frac{1}{12}. Then add the square of -\frac{1}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{6}x+\left(-\frac{1}{12}\right)^{2}=1+\left(-\frac{1}{12}\right)^{2}$$

Square -\frac{1}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{6}x+\frac{1}{144}=1+\frac{1}{144}$$

Add 1 to \frac{1}{144}.

$$x^{2}-\frac{1}{6}x+\frac{1}{144}=\frac{145}{144}$$

Factor x^{2}-\frac{1}{6}x+\frac{1}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{12}\right)^{2}=\frac{145}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{12}\right)^{2}}=\sqrt{\frac{145}{144}}$$

Simplify.

$$x-\frac{1}{12}=\frac{\sqrt{145}}{12} x-\frac{1}{12}=-\frac{\sqrt{145}}{12}$$

Add \frac{1}{12} to both sides of the equation.

$$x=\frac{\sqrt{145}+1}{12} x=\frac{1-\sqrt{145}}{12}$$