Solve 1540-112n+3n^2=0

$$1540 - 112 n + 3 n ^ { 2 } = 0$$

Solve for n

$$n=\frac{56+2\sqrt{371}i}{3}\approx 18.666666667+12.840906856i$$

$$n=\frac{-2\sqrt{371}i+56}{3}\approx 18.666666667-12.840906856i$$

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.

$$3n^{2}-112n+1540=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -112 for b, and 1540 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-112\right)±\sqrt{\left(-112\right)^{2}-4\times 3\times 1540}}{2\times 3}$$

Square -112.

$$n=\frac{-\left(-112\right)±\sqrt{12544-4\times 3\times 1540}}{2\times 3}$$

Multiply -4 times 3.

$$n=\frac{-\left(-112\right)±\sqrt{12544-12\times 1540}}{2\times 3}$$

Multiply -12 times 1540.

$$n=\frac{-\left(-112\right)±\sqrt{12544-18480}}{2\times 3}$$

Add 12544 to -18480.

$$n=\frac{-\left(-112\right)±\sqrt{-5936}}{2\times 3}$$

Take the square root of -5936.

$$n=\frac{-\left(-112\right)±4\sqrt{371}i}{2\times 3}$$

The opposite of -112 is 112.

$$n=\frac{112±4\sqrt{371}i}{2\times 3}$$

Multiply 2 times 3.

$$n=\frac{112±4\sqrt{371}i}{6}$$

Now solve the equation n=\frac{112±4\sqrt{371}i}{6} when ± is plus. Add 112 to 4i\sqrt{371}.

$$n=\frac{112+4\sqrt{371}i}{6}$$

Divide 112+4i\sqrt{371} by 6.

$$n=\frac{56+2\sqrt{371}i}{3}$$

Now solve the equation n=\frac{112±4\sqrt{371}i}{6} when ± is minus. Subtract 4i\sqrt{371} from 112.

$$n=\frac{-4\sqrt{371}i+112}{6}$$

Divide 112-4i\sqrt{371} by 6.

$$n=\frac{-2\sqrt{371}i+56}{3}$$

The equation is now solved.

$$n=\frac{56+2\sqrt{371}i}{3} n=\frac{-2\sqrt{371}i+56}{3}$$

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.

$$3n^{2}-112n+1540=0$$

Subtract 1540 from both sides of the equation.

$$3n^{2}-112n+1540-1540=-1540$$

Subtracting 1540 from itself leaves 0.

$$3n^{2}-112n=-1540$$

Divide both sides by 3.

$$\frac{3n^{2}-112n}{3}=-\frac{1540}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$n^{2}-\frac{112}{3}n=-\frac{1540}{3}$$

Divide -\frac{112}{3}, the coefficient of the x term, by 2 to get -\frac{56}{3}. Then add the square of -\frac{56}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-\frac{112}{3}n+\left(-\frac{56}{3}\right)^{2}=-\frac{1540}{3}+\left(-\frac{56}{3}\right)^{2}$$

Square -\frac{56}{3} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-\frac{112}{3}n+\frac{3136}{9}=-\frac{1540}{3}+\frac{3136}{9}$$

Add -\frac{1540}{3} to \frac{3136}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n^{2}-\frac{112}{3}n+\frac{3136}{9}=-\frac{1484}{9}$$

Factor n^{2}-\frac{112}{3}n+\frac{3136}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-\frac{56}{3}\right)^{2}=-\frac{1484}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{56}{3}\right)^{2}}=\sqrt{-\frac{1484}{9}}$$

Simplify.

$$n-\frac{56}{3}=\frac{2\sqrt{371}i}{3} n-\frac{56}{3}=-\frac{2\sqrt{371}i}{3}$$

Add \frac{56}{3} to both sides of the equation.

$$n=\frac{56+2\sqrt{371}i}{3} n=\frac{-2\sqrt{371}i+56}{3}$$