Solve n^2+30.1258n-12050.32=0

$$n ^ { 2 } + 30.1258 n - 12050.32 = 0$$

Solve for n

$$n = \frac{\sqrt{1227721095641} - 150629}{10000} \approx 95.739676488$$

$$n=\frac{-\sqrt{1227721095641}-150629}{10000}\approx -125.865476488$$

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.

$$n^{2}+30.1258n-12050.32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 30.1258 for b, and -12050.32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-30.1258±\sqrt{30.1258^{2}-4\left(-12050.32\right)}}{2}$$

Square 30.1258 by squaring both the numerator and the denominator of the fraction.

$$n=\frac{-30.1258±\sqrt{907.56382564-4\left(-12050.32\right)}}{2}$$

Multiply -4 times -12050.32.

$$n=\frac{-30.1258±\sqrt{907.56382564+48201.28}}{2}$$

Add 907.56382564 to 48201.28 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n=\frac{-30.1258±\sqrt{49108.84382564}}{2}$$

Take the square root of 49108.84382564.

$$n=\frac{-30.1258±\frac{\sqrt{1227721095641}}{5000}}{2}$$

Now solve the equation n=\frac{-30.1258±\frac{\sqrt{1227721095641}}{5000}}{2} when ± is plus. Add -30.1258 to \frac{\sqrt{1227721095641}}{5000}.

$$n=\frac{\sqrt{1227721095641}-150629}{2\times 5000}$$

Divide \frac{-150629+\sqrt{1227721095641}}{5000} by 2.

$$n=\frac{\sqrt{1227721095641}-150629}{10000}$$

Now solve the equation n=\frac{-30.1258±\frac{\sqrt{1227721095641}}{5000}}{2} when ± is minus. Subtract \frac{\sqrt{1227721095641}}{5000} from -30.1258.

$$n=\frac{-\sqrt{1227721095641}-150629}{2\times 5000}$$

Divide \frac{-150629-\sqrt{1227721095641}}{5000} by 2.

$$n=\frac{-\sqrt{1227721095641}-150629}{10000}$$

The equation is now solved.

$$n=\frac{\sqrt{1227721095641}-150629}{10000} n=\frac{-\sqrt{1227721095641}-150629}{10000}$$

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.

$$n^{2}+30.1258n-12050.32=0$$

Add 12050.32 to both sides of the equation.

$$n^{2}+30.1258n-12050.32-\left(-12050.32\right)=-\left(-12050.32\right)$$

Subtracting -12050.32 from itself leaves 0.

$$n^{2}+30.1258n=-\left(-12050.32\right)$$

Subtract -12050.32 from 0.

$$n^{2}+30.1258n=12050.32$$

Divide 30.1258, the coefficient of the x term, by 2 to get 15.0629. Then add the square of 15.0629 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}+30.1258n+15.0629^{2}=12050.32+15.0629^{2}$$

Square 15.0629 by squaring both the numerator and the denominator of the fraction.

$$n^{2}+30.1258n+226.89095641=12050.32+226.89095641$$

Add 12050.32 to 226.89095641 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n^{2}+30.1258n+226.89095641=12277.21095641$$

Factor n^{2}+30.1258n+226.89095641. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n+15.0629\right)^{2}=12277.21095641$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+15.0629\right)^{2}}=\sqrt{12277.21095641}$$

Simplify.

$$n+15.0629=\frac{\sqrt{1227721095641}}{10000} n+15.0629=-\frac{\sqrt{1227721095641}}{10000}$$

Subtract 15.0629 from both sides of the equation.

$$n=\frac{\sqrt{1227721095641}-150629}{10000} n=\frac{-\sqrt{1227721095641}-150629}{10000}$$