Solve 61n-3n^2=10620

$$61 n - 3 n ^ { 2 } = 10620$$

Solve for n

$$n=\frac{-\sqrt{123719}i+61}{6}\approx 10.166666667-58.622852958i$$

$$n=\frac{61+\sqrt{123719}i}{6}\approx 10.166666667+58.622852958i$$

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.

$$-3n^{2}+61n=10620$$

Subtract 10620 from both sides of the equation.

$$-3n^{2}+61n-10620=10620-10620$$

Subtracting 10620 from itself leaves 0.

$$-3n^{2}+61n-10620=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 61 for b, and -10620 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-61±\sqrt{61^{2}-4\left(-3\right)\left(-10620\right)}}{2\left(-3\right)}$$

Square 61.

$$n=\frac{-61±\sqrt{3721-4\left(-3\right)\left(-10620\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$n=\frac{-61±\sqrt{3721+12\left(-10620\right)}}{2\left(-3\right)}$$

Multiply 12 times -10620.

$$n=\frac{-61±\sqrt{3721-127440}}{2\left(-3\right)}$$

Add 3721 to -127440.

$$n=\frac{-61±\sqrt{-123719}}{2\left(-3\right)}$$

Take the square root of -123719.

$$n=\frac{-61±\sqrt{123719}i}{2\left(-3\right)}$$

Multiply 2 times -3.

$$n=\frac{-61±\sqrt{123719}i}{-6}$$

Now solve the equation n=\frac{-61±\sqrt{123719}i}{-6} when ± is plus. Add -61 to i\sqrt{123719}.

$$n=\frac{-61+\sqrt{123719}i}{-6}$$

Divide -61+i\sqrt{123719} by -6.

$$n=\frac{-\sqrt{123719}i+61}{6}$$

Now solve the equation n=\frac{-61±\sqrt{123719}i}{-6} when ± is minus. Subtract i\sqrt{123719} from -61.

$$n=\frac{-\sqrt{123719}i-61}{-6}$$

Divide -61-i\sqrt{123719} by -6.

$$n=\frac{61+\sqrt{123719}i}{6}$$

The equation is now solved.

$$n=\frac{-\sqrt{123719}i+61}{6} n=\frac{61+\sqrt{123719}i}{6}$$

Steps for Completing the Square

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.

$$-3n^{2}+61n=10620$$

Divide both sides by -3.

$$\frac{-3n^{2}+61n}{-3}=\frac{10620}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$n^{2}+\frac{61}{-3}n=\frac{10620}{-3}$$

Divide 61 by -3.

$$n^{2}-\frac{61}{3}n=\frac{10620}{-3}$$

Divide 10620 by -3.

$$n^{2}-\frac{61}{3}n=-3540$$

Divide -\frac{61}{3}, the coefficient of the x term, by 2 to get -\frac{61}{6}. Then add the square of -\frac{61}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-\frac{61}{3}n+\left(-\frac{61}{6}\right)^{2}=-3540+\left(-\frac{61}{6}\right)^{2}$$

Square -\frac{61}{6} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-\frac{61}{3}n+\frac{3721}{36}=-3540+\frac{3721}{36}$$

Add -3540 to \frac{3721}{36}.

$$n^{2}-\frac{61}{3}n+\frac{3721}{36}=-\frac{123719}{36}$$

Factor n^{2}-\frac{61}{3}n+\frac{3721}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-\frac{61}{6}\right)^{2}=-\frac{123719}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{61}{6}\right)^{2}}=\sqrt{-\frac{123719}{36}}$$

Simplify.

$$n-\frac{61}{6}=\frac{\sqrt{123719}i}{6} n-\frac{61}{6}=-\frac{\sqrt{123719}i}{6}$$

Add \frac{61}{6} to both sides of the equation.

$$n=\frac{61+\sqrt{123719}i}{6} n=\frac{-\sqrt{123719}i+61}{6}$$