Solve 3n^2=9n

$$3 n ^ { 2 } = 9 n$$

Solve for n

$$n=3$$

$$n=0$$

Steps Using Factoring

Subtract 9n from both sides.

$$3n^{2}-9n=0$$

Factor out n.

$$n\left(3n-9\right)=0$$

To find equation solutions, solve n=0 and 3n-9=0.

$$n=0 n=3$$

Steps Using the Quadratic Formula

Subtract 9n from both sides.

$$3n^{2}-9n=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -9 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}}}{2\times 3}$$

Take the square root of \left(-9\right)^{2}.

$$n=\frac{-\left(-9\right)±9}{2\times 3}$$

The opposite of -9 is 9.

$$n=\frac{9±9}{2\times 3}$$

Multiply 2 times 3.

$$n=\frac{9±9}{6}$$

Now solve the equation n=\frac{9±9}{6} when ± is plus. Add 9 to 9.

$$n=\frac{18}{6}$$

Divide 18 by 6.

$$n=3$$

Now solve the equation n=\frac{9±9}{6} when ± is minus. Subtract 9 from 9.

$$n=\frac{0}{6}$$

Divide 0 by 6.

$$n=0$$

The equation is now solved.

$$n=3 n=0$$

Steps for Completing the Square

Subtract 9n from both sides.

$$3n^{2}-9n=0$$

Divide both sides by 3.

$$\frac{3n^{2}-9n}{3}=\frac{0}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$n^{2}+\left(-\frac{9}{3}\right)n=\frac{0}{3}$$

Divide -9 by 3.

$$n^{2}-3n=\frac{0}{3}$$

Divide 0 by 3.

$$n^{2}-3n=0$$

Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-3n+\left(-\frac{3}{2}\right)^{2}=\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-3n+\frac{9}{4}=\frac{9}{4}$$

Factor n^{2}-3n+\frac{9}{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}.

$$\left(n-\frac{3}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{3}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

$$n-\frac{3}{2}=\frac{3}{2} n-\frac{3}{2}=-\frac{3}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$n=3 n=0$$