Solve x/x-x-1/3=0.6

$$\frac { x } { x } - \frac { x - 1 } { 3 } = 0.6$$

Solve for x

$$x=2.2$$

Steps Using Factoring

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 3x, the least common multiple of x,3.

$$3x-x\left(x-1\right)=1.8x$$

Use the distributive property to multiply x by x-1.

$$3x-\left(x^{2}-x\right)=1.8x$$

To find the opposite of x^{2}-x, find the opposite of each term.

$$3x-x^{2}-\left(-x\right)=1.8x$$

The opposite of -x is x.

$$3x-x^{2}+x=1.8x$$

Combine 3x and x to get 4x.

$$4x-x^{2}=1.8x$$

Subtract 1.8x from both sides.

$$4x-x^{2}-1.8x=0$$

Combine 4x and -1.8x to get 2.2x.

$$2.2x-x^{2}=0$$

Factor out x.

$$x\left(2.2-x\right)=0$$

To find equation solutions, solve x=0 and 2.2-x=0.

$$x=0 x=\frac{11}{5}$$

Variable x cannot be equal to 0.

$$x=\frac{11}{5}$$

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 3x, the least common multiple of x,3.

$$3x-x\left(x-1\right)=1.8x$$

Use the distributive property to multiply x by x-1.

$$3x-\left(x^{2}-x\right)=1.8x$$

To find the opposite of x^{2}-x, find the opposite of each term.

$$3x-x^{2}-\left(-x\right)=1.8x$$

The opposite of -x is x.

$$3x-x^{2}+x=1.8x$$

Combine 3x and x to get 4x.

$$4x-x^{2}=1.8x$$

Subtract 1.8x from both sides.

$$4x-x^{2}-1.8x=0$$

Combine 4x and -1.8x to get 2.2x.

$$2.2x-x^{2}=0$$

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.

$$-x^{2}+\frac{11}{5}x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, \frac{11}{5} for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{11}{5}±\sqrt{\left(\frac{11}{5}\right)^{2}}}{2\left(-1\right)}$$

Take the square root of \left(\frac{11}{5}\right)^{2}.

$$x=\frac{-\frac{11}{5}±\frac{11}{5}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-\frac{11}{5}±\frac{11}{5}}{-2}$$

Now solve the equation x=\frac{-\frac{11}{5}±\frac{11}{5}}{-2} when ± is plus. Add -\frac{11}{5} to \frac{11}{5} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

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

Divide 0 by -2.

$$x=0$$

Now solve the equation x=\frac{-\frac{11}{5}±\frac{11}{5}}{-2} when ± is minus. Subtract \frac{11}{5} from -\frac{11}{5} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{\frac{22}{5}}{-2}$$

Divide -\frac{22}{5} by -2.

$$x=\frac{11}{5}$$

The equation is now solved.

$$x=0 x=\frac{11}{5}$$

Variable x cannot be equal to 0.

$$x=\frac{11}{5}$$

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 3x, the least common multiple of x,3.

$$3x-x\left(x-1\right)=1.8x$$

Use the distributive property to multiply x by x-1.

$$3x-\left(x^{2}-x\right)=1.8x$$

To find the opposite of x^{2}-x, find the opposite of each term.

$$3x-x^{2}-\left(-x\right)=1.8x$$

The opposite of -x is x.

$$3x-x^{2}+x=1.8x$$

Combine 3x and x to get 4x.

$$4x-x^{2}=1.8x$$

Subtract 1.8x from both sides.

$$4x-x^{2}-1.8x=0$$

Combine 4x and -1.8x to get 2.2x.

$$2.2x-x^{2}=0$$

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.

$$-x^{2}+\frac{11}{5}x=0$$

Divide both sides by -1.

$$\frac{-x^{2}+\frac{11}{5}x}{-1}=\frac{0}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{\frac{11}{5}}{-1}x=\frac{0}{-1}$$

Divide \frac{11}{5} by -1.

$$x^{2}-\frac{11}{5}x=\frac{0}{-1}$$

Divide 0 by -1.

$$x^{2}-\frac{11}{5}x=0$$

Divide -\frac{11}{5}, the coefficient of the x term, by 2 to get -\frac{11}{10}. Then add the square of -\frac{11}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{11}{5}x+\left(-\frac{11}{10}\right)^{2}=\left(-\frac{11}{10}\right)^{2}$$

Square -\frac{11}{10} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{11}{5}x+\frac{121}{100}=\frac{121}{100}$$

Factor x^{2}-\frac{11}{5}x+\frac{121}{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}.

$$\left(x-\frac{11}{10}\right)^{2}=\frac{121}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{10}\right)^{2}}=\sqrt{\frac{121}{100}}$$

Simplify.

$$x-\frac{11}{10}=\frac{11}{10} x-\frac{11}{10}=-\frac{11}{10}$$

Add \frac{11}{10} to both sides of the equation.

$$x=\frac{11}{5} x=0$$

Variable x cannot be equal to 0.

$$x=\frac{11}{5}$$