Solve n^2-4019n+2009^2=0

$$n ^ { 2 } - 4019 n + 2009 ^ { 2 } = 0$$

Solve for n

$$n = \frac{3 \sqrt{893} + 4019}{2} \approx 2054.324658392$$

$$n = \frac{4019 - 3 \sqrt{893}}{2} \approx 1964.675341608$$

Steps Using the Quadratic Formula

Calculate 2009 to the power of 2 and get 4036081.

$$n^{2}-4019n+4036081=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -4019 for b, and 4036081 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-4019\right)±\sqrt{\left(-4019\right)^{2}-4\times 4036081}}{2}$$

Square -4019.

$$n=\frac{-\left(-4019\right)±\sqrt{16152361-4\times 4036081}}{2}$$

Multiply -4 times 4036081.

$$n=\frac{-\left(-4019\right)±\sqrt{16152361-16144324}}{2}$$

Add 16152361 to -16144324.

$$n=\frac{-\left(-4019\right)±\sqrt{8037}}{2}$$

Take the square root of 8037.

$$n=\frac{-\left(-4019\right)±3\sqrt{893}}{2}$$

The opposite of -4019 is 4019.

$$n=\frac{4019±3\sqrt{893}}{2}$$

Now solve the equation n=\frac{4019±3\sqrt{893}}{2} when ± is plus. Add 4019 to 3\sqrt{893}.

$$n=\frac{3\sqrt{893}+4019}{2}$$

Now solve the equation n=\frac{4019±3\sqrt{893}}{2} when ± is minus. Subtract 3\sqrt{893} from 4019.

$$n=\frac{4019-3\sqrt{893}}{2}$$

The equation is now solved.

$$n=\frac{3\sqrt{893}+4019}{2} n=\frac{4019-3\sqrt{893}}{2}$$

Steps for Completing the Square

Calculate 2009 to the power of 2 and get 4036081.

$$n^{2}-4019n+4036081=0$$

Subtract 4036081 from both sides. Anything subtracted from zero gives its negation.

$$n^{2}-4019n=-4036081$$

Divide -4019, the coefficient of the x term, by 2 to get -\frac{4019}{2}. Then add the square of -\frac{4019}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-4019n+\left(-\frac{4019}{2}\right)^{2}=-4036081+\left(-\frac{4019}{2}\right)^{2}$$

Square -\frac{4019}{2} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-4019n+\frac{16152361}{4}=-4036081+\frac{16152361}{4}$$

Add -4036081 to \frac{16152361}{4}.

$$n^{2}-4019n+\frac{16152361}{4}=\frac{8037}{4}$$

Factor n^{2}-4019n+\frac{16152361}{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}.

$$\left(n-\frac{4019}{2}\right)^{2}=\frac{8037}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{4019}{2}\right)^{2}}=\sqrt{\frac{8037}{4}}$$

Simplify.

$$n-\frac{4019}{2}=\frac{3\sqrt{893}}{2} n-\frac{4019}{2}=-\frac{3\sqrt{893}}{2}$$

Add \frac{4019}{2} to both sides of the equation.

$$n=\frac{3\sqrt{893}+4019}{2} n=\frac{4019-3\sqrt{893}}{2}$$