Solve 6x-5=1/x

$$6x-5= \frac{ 1 }{ x }$$

Solve for x

$$x=-\frac{1}{6}\approx -0.166666667$$

$$x=1$$

Steps Using the Quadratic Formula

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

$$6xx+x\left(-5\right)=1$$

Multiply x and x to get x^{2}.

$$6x^{2}+x\left(-5\right)=1$$

Subtract 1 from both sides.

$$6x^{2}+x\left(-5\right)-1=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$6x^{2}-5x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -5 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 6\left(-1\right)}}{2\times 6}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 6\left(-1\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-5\right)±\sqrt{25-24\left(-1\right)}}{2\times 6}$$

Multiply -24 times -1.

$$x=\frac{-\left(-5\right)±\sqrt{25+24}}{2\times 6}$$

Add 25 to 24.

$$x=\frac{-\left(-5\right)±\sqrt{49}}{2\times 6}$$

Take the square root of 49.

$$x=\frac{-\left(-5\right)±7}{2\times 6}$$

The opposite of -5 is 5.

$$x=\frac{5±7}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{5±7}{12}$$

Now solve the equation x=\frac{5±7}{12} when ± is plus. Add 5 to 7.

$$x=\frac{12}{12}$$

Divide 12 by 12.

$$x=1$$

Now solve the equation x=\frac{5±7}{12} when ± is minus. Subtract 7 from 5.

$$x=-\frac{2}{12}$$

Reduce the fraction \frac{-2}{12} to lowest terms by extracting and canceling out 2.

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

The equation is now solved.

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

Steps for Completing the Square

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

$$6xx+x\left(-5\right)=1$$

Multiply x and x to get x^{2}.

$$6x^{2}+x\left(-5\right)=1$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$6x^{2}-5x=1$$

Divide both sides by 6.

$$\frac{6x^{2}-5x}{6}=\frac{1}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}-\frac{5}{6}x=\frac{1}{6}$$

Divide -\frac{5}{6}, the coefficient of the x term, by 2 to get -\frac{5}{12}. Then add the square of -\frac{5}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{6}x+\left(-\frac{5}{12}\right)^{2}=\frac{1}{6}+\left(-\frac{5}{12}\right)^{2}$$

Square -\frac{5}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{6}x+\frac{25}{144}=\frac{1}{6}+\frac{25}{144}$$

Add \frac{1}{6} to \frac{25}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{6}x+\frac{25}{144}=\frac{49}{144}$$

Factor x^{2}-\frac{5}{6}x+\frac{25}{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}.

$$\left(x-\frac{5}{12}\right)^{2}=\frac{49}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{12}\right)^{2}}=\sqrt{\frac{49}{144}}$$

Simplify.

$$x-\frac{5}{12}=\frac{7}{12} x-\frac{5}{12}=-\frac{7}{12}$$

Add \frac{5}{12} to both sides of the equation.

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