Solve x-1+1/x=6(x+0)

$$x - 1 + \frac { 1 } { x } = 6 ( x + 0 )$$

Solve for x

$$x=\frac{\sqrt{21}-1}{10}\approx 0.358257569$$

$$x=\frac{-\sqrt{21}-1}{10}\approx -0.558257569$$

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.

$$xx+x\left(-1\right)+1=6\left(x+0\right)x$$

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

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

Anything plus zero gives itself.

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

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

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

Subtract 6x^{2} from both sides.

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

Combine x^{2} and -6x^{2} to get -5x^{2}.

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

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.

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

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

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

Multiply -4 times -5.

$$x=\frac{-\left(-1\right)±\sqrt{1+20}}{2\left(-5\right)}$$

Add 1 to 20.

$$x=\frac{-\left(-1\right)±\sqrt{21}}{2\left(-5\right)}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{21}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{1±\sqrt{21}}{-10}$$

Now solve the equation x=\frac{1±\sqrt{21}}{-10} when ± is plus. Add 1 to \sqrt{21}.

$$x=\frac{\sqrt{21}+1}{-10}$$

Divide 1+\sqrt{21} by -10.

$$x=\frac{-\sqrt{21}-1}{10}$$

Now solve the equation x=\frac{1±\sqrt{21}}{-10} when ± is minus. Subtract \sqrt{21} from 1.

$$x=\frac{1-\sqrt{21}}{-10}$$

Divide 1-\sqrt{21} by -10.

$$x=\frac{\sqrt{21}-1}{10}$$

The equation is now solved.

$$x=\frac{-\sqrt{21}-1}{10} x=\frac{\sqrt{21}-1}{10}$$

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.

$$xx+x\left(-1\right)+1=6\left(x+0\right)x$$

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

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

Anything plus zero gives itself.

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

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

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

Subtract 6x^{2} from both sides.

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

Combine x^{2} and -6x^{2} to get -5x^{2}.

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

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

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

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.

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

Divide both sides by -5.

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

Dividing by -5 undoes the multiplication by -5.

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

Divide -1 by -5.

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

Divide -1 by -5.

$$x^{2}+\frac{1}{5}x=\frac{1}{5}$$

Divide \frac{1}{5}, the coefficient of the x term, by 2 to get \frac{1}{10}. Then add the square of \frac{1}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{5}x+\left(\frac{1}{10}\right)^{2}=\frac{1}{5}+\left(\frac{1}{10}\right)^{2}$$

Square \frac{1}{10} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{5}x+\frac{1}{100}=\frac{1}{5}+\frac{1}{100}$$

Add \frac{1}{5} to \frac{1}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{5}x+\frac{1}{100}=\frac{21}{100}$$

Factor x^{2}+\frac{1}{5}x+\frac{1}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{10}\right)^{2}=\frac{21}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{10}\right)^{2}}=\sqrt{\frac{21}{100}}$$

Simplify.

$$x+\frac{1}{10}=\frac{\sqrt{21}}{10} x+\frac{1}{10}=-\frac{\sqrt{21}}{10}$$

Subtract \frac{1}{10} from both sides of the equation.

$$x=\frac{\sqrt{21}-1}{10} x=\frac{-\sqrt{21}-1}{10}$$