Solve 9x-1/x=1

$$9 x - \frac { 1 } { x } = 1$$

Solve for x

$$x=\frac{\sqrt{37}+1}{18}\approx 0.393486807$$

$$x=\frac{1-\sqrt{37}}{18}\approx -0.282375696$$

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.

$$9xx-1=x$$

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

$$9x^{2}-1=x$$

Subtract x from both sides.

$$9x^{2}-1-x=0$$

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.

$$9x^{2}-x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, -1 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 9\left(-1\right)}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-1\right)±\sqrt{1-36\left(-1\right)}}{2\times 9}$$

Multiply -36 times -1.

$$x=\frac{-\left(-1\right)±\sqrt{1+36}}{2\times 9}$$

Add 1 to 36.

$$x=\frac{-\left(-1\right)±\sqrt{37}}{2\times 9}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{37}}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{1±\sqrt{37}}{18}$$

Now solve the equation x=\frac{1±\sqrt{37}}{18} when ± is plus. Add 1 to \sqrt{37}.

$$x=\frac{\sqrt{37}+1}{18}$$

Now solve the equation x=\frac{1±\sqrt{37}}{18} when ± is minus. Subtract \sqrt{37} from 1.

$$x=\frac{1-\sqrt{37}}{18}$$

The equation is now solved.

$$x=\frac{\sqrt{37}+1}{18} x=\frac{1-\sqrt{37}}{18}$$

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.

$$9xx-1=x$$

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

$$9x^{2}-1=x$$

Subtract x from both sides.

$$9x^{2}-1-x=0$$

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

$$9x^{2}-x=1$$

Divide both sides by 9.

$$\frac{9x^{2}-x}{9}=\frac{1}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$x^{2}-\frac{1}{9}x=\frac{1}{9}$$

Divide -\frac{1}{9}, the coefficient of the x term, by 2 to get -\frac{1}{18}. Then add the square of -\frac{1}{18} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{9}x+\left(-\frac{1}{18}\right)^{2}=\frac{1}{9}+\left(-\frac{1}{18}\right)^{2}$$

Square -\frac{1}{18} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{9}x+\frac{1}{324}=\frac{1}{9}+\frac{1}{324}$$

Add \frac{1}{9} to \frac{1}{324} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{9}x+\frac{1}{324}=\frac{37}{324}$$

Factor x^{2}-\frac{1}{9}x+\frac{1}{324}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{18}\right)^{2}=\frac{37}{324}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{18}\right)^{2}}=\sqrt{\frac{37}{324}}$$

Simplify.

$$x-\frac{1}{18}=\frac{\sqrt{37}}{18} x-\frac{1}{18}=-\frac{\sqrt{37}}{18}$$

Add \frac{1}{18} to both sides of the equation.

$$x=\frac{\sqrt{37}+1}{18} x=\frac{1-\sqrt{37}}{18}$$