Solve x^2/9-x=0

$$\frac{ { x }^{ 2 } }{ 9 } -x=0$$

Solve for x

$$x=9$$

$$x=0$$

Steps Using Factoring

Multiply both sides of the equation by 9.

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

Factor out x.

$$x\left(x-9\right)=0$$

To find equation solutions, solve x=0 and x-9=0.

$$x=0 x=9$$

Steps Using the Quadratic Formula

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.

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

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

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

Take the square root of 1.

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

The opposite of -1 is 1.

$$x=\frac{1±1}{2\times \frac{1}{9}}$$

Multiply 2 times \frac{1}{9}.

$$x=\frac{1±1}{\frac{2}{9}}$$

Now solve the equation x=\frac{1±1}{\frac{2}{9}} when ± is plus. Add 1 to 1.

$$x=\frac{2}{\frac{2}{9}}$$

Divide 2 by \frac{2}{9} by multiplying 2 by the reciprocal of \frac{2}{9}.

$$x=9$$

Now solve the equation x=\frac{1±1}{\frac{2}{9}} when ± is minus. Subtract 1 from 1.

$$x=\frac{0}{\frac{2}{9}}$$

Divide 0 by \frac{2}{9} by multiplying 0 by the reciprocal of \frac{2}{9}.

$$x=0$$

The equation is now solved.

$$x=9 x=0$$

Steps for Completing the Square

Multiply both sides of the equation by 9.

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

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

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

Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-9x+\frac{81}{4}=\frac{81}{4}$$

Factor x^{2}-9x+\frac{81}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{81}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{81}{4}}$$

Simplify.

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

Add \frac{9}{2} to both sides of the equation.

$$x=9 x=0$$