Solve -n^2=11n

$$- n ^ { 2 } = 11 n$$

Solve for n

$$n=-11$$

$$n=0$$

Steps Using the Quadratic Formula

Subtract 11n from both sides.

$$-n^{2}-11n=0$$

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

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

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

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

The opposite of -11 is 11.

$$n=\frac{11±11}{2\left(-1\right)}$$

Multiply 2 times -1.

$$n=\frac{11±11}{-2}$$

Now solve the equation n=\frac{11±11}{-2} when ± is plus. Add 11 to 11.

$$n=\frac{22}{-2}$$

Divide 22 by -2.

$$n=-11$$

Now solve the equation n=\frac{11±11}{-2} when ± is minus. Subtract 11 from 11.

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

Divide 0 by -2.

$$n=0$$

The equation is now solved.

$$n=-11 n=0$$

Steps for Completing the Square

Subtract 11n from both sides.

$$-n^{2}-11n=0$$

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

$$n^{2}+\left(-\frac{11}{-1}\right)n=\frac{0}{-1}$$

Divide -11 by -1.

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

Divide 0 by -1.

$$n^{2}+11n=0$$

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

$$n^{2}+11n+\left(\frac{11}{2}\right)^{2}=\left(\frac{11}{2}\right)^{2}$$

Square \frac{11}{2} by squaring both the numerator and the denominator of the fraction.

$$n^{2}+11n+\frac{121}{4}=\frac{121}{4}$$

Factor n^{2}+11n+\frac{121}{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}.

$$\left(n+\frac{11}{2}\right)^{2}=\frac{121}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+\frac{11}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Simplify.

$$n+\frac{11}{2}=\frac{11}{2} n+\frac{11}{2}=-\frac{11}{2}$$

Subtract \frac{11}{2} from both sides of the equation.

$$n=0 n=-11$$