Solve n^2-121n+7140=0

$${ n }^{ 2 } -121n+7140=0$$

Solve for n

$$n=\frac{121+\sqrt{13919}i}{2}\approx 60.5+58.989405829i$$

$$n=\frac{-\sqrt{13919}i+121}{2}\approx 60.5-58.989405829i$$

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.

$$n^{2}-121n+7140=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -121 for b, and 7140 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-121\right)±\sqrt{\left(-121\right)^{2}-4\times 7140}}{2}$$

Square -121.

$$n=\frac{-\left(-121\right)±\sqrt{14641-4\times 7140}}{2}$$

Multiply -4 times 7140.

$$n=\frac{-\left(-121\right)±\sqrt{14641-28560}}{2}$$

Add 14641 to -28560.

$$n=\frac{-\left(-121\right)±\sqrt{-13919}}{2}$$

Take the square root of -13919.

$$n=\frac{-\left(-121\right)±\sqrt{13919}i}{2}$$

The opposite of -121 is 121.

$$n=\frac{121±\sqrt{13919}i}{2}$$

Now solve the equation n=\frac{121±\sqrt{13919}i}{2} when ± is plus. Add 121 to i\sqrt{13919}.

$$n=\frac{121+\sqrt{13919}i}{2}$$

Now solve the equation n=\frac{121±\sqrt{13919}i}{2} when ± is minus. Subtract i\sqrt{13919} from 121.

$$n=\frac{-\sqrt{13919}i+121}{2}$$

The equation is now solved.

$$n=\frac{121+\sqrt{13919}i}{2} n=\frac{-\sqrt{13919}i+121}{2}$$

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.

$$n^{2}-121n+7140=0$$

Subtract 7140 from both sides of the equation.

$$n^{2}-121n+7140-7140=-7140$$

Subtracting 7140 from itself leaves 0.

$$n^{2}-121n=-7140$$

Divide -121, the coefficient of the x term, by 2 to get -\frac{121}{2}. Then add the square of -\frac{121}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-121n+\left(-\frac{121}{2}\right)^{2}=-7140+\left(-\frac{121}{2}\right)^{2}$$

Square -\frac{121}{2} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-121n+\frac{14641}{4}=-7140+\frac{14641}{4}$$

Add -7140 to \frac{14641}{4}.

$$n^{2}-121n+\frac{14641}{4}=-\frac{13919}{4}$$

Factor n^{2}-121n+\frac{14641}{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}.

$$\left(n-\frac{121}{2}\right)^{2}=-\frac{13919}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{121}{2}\right)^{2}}=\sqrt{-\frac{13919}{4}}$$

Simplify.

$$n-\frac{121}{2}=\frac{\sqrt{13919}i}{2} n-\frac{121}{2}=-\frac{\sqrt{13919}i}{2}$$

Add \frac{121}{2} to both sides of the equation.

$$n=\frac{121+\sqrt{13919}i}{2} n=\frac{-\sqrt{13919}i+121}{2}$$