Solve a=3a/a-1

$$a = \frac { 3 a } { a - 1 }$$

Solve for a

$$a=4$$

$$a=0$$

Steps Using Factoring

Subtract \frac{3a}{a-1} from both sides.

$$a-\frac{3a}{a-1}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply a times \frac{a-1}{a-1}.

$$\frac{a\left(a-1\right)}{a-1}-\frac{3a}{a-1}=0$$

Since \frac{a\left(a-1\right)}{a-1} and \frac{3a}{a-1} have the same denominator, subtract them by subtracting their numerators.

$$\frac{a\left(a-1\right)-3a}{a-1}=0$$

Do the multiplications in a\left(a-1\right)-3a.

$$\frac{a^{2}-a-3a}{a-1}=0$$

Combine like terms in a^{2}-a-3a.

$$\frac{a^{2}-4a}{a-1}=0$$

Variable a cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by a-1.

$$a^{2}-4a=0$$

Factor out a.

$$a\left(a-4\right)=0$$

To find equation solutions, solve a=0 and a-4=0.

$$a=0 a=4$$

Steps Using the Quadratic Formula

Subtract \frac{3a}{a-1} from both sides.

$$a-\frac{3a}{a-1}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply a times \frac{a-1}{a-1}.

$$\frac{a\left(a-1\right)}{a-1}-\frac{3a}{a-1}=0$$

Since \frac{a\left(a-1\right)}{a-1} and \frac{3a}{a-1} have the same denominator, subtract them by subtracting their numerators.

$$\frac{a\left(a-1\right)-3a}{a-1}=0$$

Do the multiplications in a\left(a-1\right)-3a.

$$\frac{a^{2}-a-3a}{a-1}=0$$

Combine like terms in a^{2}-a-3a.

$$\frac{a^{2}-4a}{a-1}=0$$

Variable a cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by a-1.

$$a^{2}-4a=0$$

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

$$a=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}}}{2}$$

Take the square root of \left(-4\right)^{2}.

$$a=\frac{-\left(-4\right)±4}{2}$$

The opposite of -4 is 4.

$$a=\frac{4±4}{2}$$

Now solve the equation a=\frac{4±4}{2} when ± is plus. Add 4 to 4.

$$a=\frac{8}{2}$$

Divide 8 by 2.

$$a=4$$

Now solve the equation a=\frac{4±4}{2} when ± is minus. Subtract 4 from 4.

$$a=\frac{0}{2}$$

Divide 0 by 2.

$$a=0$$

The equation is now solved.

$$a=4 a=0$$

Steps for Completing the Square

Subtract \frac{3a}{a-1} from both sides.

$$a-\frac{3a}{a-1}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply a times \frac{a-1}{a-1}.

$$\frac{a\left(a-1\right)}{a-1}-\frac{3a}{a-1}=0$$

Since \frac{a\left(a-1\right)}{a-1} and \frac{3a}{a-1} have the same denominator, subtract them by subtracting their numerators.

$$\frac{a\left(a-1\right)-3a}{a-1}=0$$

Do the multiplications in a\left(a-1\right)-3a.

$$\frac{a^{2}-a-3a}{a-1}=0$$

Combine like terms in a^{2}-a-3a.

$$\frac{a^{2}-4a}{a-1}=0$$

Variable a cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by a-1.

$$a^{2}-4a=0$$

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-4a+\left(-2\right)^{2}=\left(-2\right)^{2}$$

Square -2.

$$a^{2}-4a+4=4$$

Factor a^{2}-4a+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}.

$$\left(a-2\right)^{2}=4$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-2\right)^{2}}=\sqrt{4}$$

Simplify.

$$a-2=2 a-2=-2$$

Add 2 to both sides of the equation.

$$a=4 a=0$$