Solve 666*2-24=n(n-1)/2

$$666 \times 2 - 24 = \frac { n ( n - 1 ) } { 2 }$$

Solve for n

$$n = \frac{\sqrt{10465} + 1}{2} \approx 51.649291295$$

$$n=\frac{1-\sqrt{10465}}{2}\approx -50.649291295$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$1332\times 2-48=n\left(n-1\right)$$

Multiply 1332 and 2 to get 2664.

$$2664-48=n\left(n-1\right)$$

Subtract 48 from 2664 to get 2616.

$$2616=n\left(n-1\right)$$

Use the distributive property to multiply n by n-1.

$$2616=n^{2}-n$$

Swap sides so that all variable terms are on the left hand side.

$$n^{2}-n=2616$$

Subtract 2616 from both sides.

$$n^{2}-n-2616=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1 for b, and -2616 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-1\right)±\sqrt{1-4\left(-2616\right)}}{2}$$

Multiply -4 times -2616.

$$n=\frac{-\left(-1\right)±\sqrt{1+10464}}{2}$$

Add 1 to 10464.

$$n=\frac{-\left(-1\right)±\sqrt{10465}}{2}$$

The opposite of -1 is 1.

$$n=\frac{1±\sqrt{10465}}{2}$$

Now solve the equation n=\frac{1±\sqrt{10465}}{2} when ± is plus. Add 1 to \sqrt{10465}.

$$n=\frac{\sqrt{10465}+1}{2}$$

Now solve the equation n=\frac{1±\sqrt{10465}}{2} when ± is minus. Subtract \sqrt{10465} from 1.

$$n=\frac{1-\sqrt{10465}}{2}$$

The equation is now solved.

$$n=\frac{\sqrt{10465}+1}{2} n=\frac{1-\sqrt{10465}}{2}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$1332\times 2-48=n\left(n-1\right)$$

Multiply 1332 and 2 to get 2664.

$$2664-48=n\left(n-1\right)$$

Subtract 48 from 2664 to get 2616.

$$2616=n\left(n-1\right)$$

Use the distributive property to multiply n by n-1.

$$2616=n^{2}-n$$

Swap sides so that all variable terms are on the left hand side.

$$n^{2}-n=2616$$

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-n+\left(-\frac{1}{2}\right)^{2}=2616+\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-n+\frac{1}{4}=2616+\frac{1}{4}$$

Add 2616 to \frac{1}{4}.

$$n^{2}-n+\frac{1}{4}=\frac{10465}{4}$$

Factor n^{2}-n+\frac{1}{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}.

$$\left(n-\frac{1}{2}\right)^{2}=\frac{10465}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{1}{2}\right)^{2}}=\sqrt{\frac{10465}{4}}$$

Simplify.

$$n-\frac{1}{2}=\frac{\sqrt{10465}}{2} n-\frac{1}{2}=-\frac{\sqrt{10465}}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$n=\frac{\sqrt{10465}+1}{2} n=\frac{1-\sqrt{10465}}{2}$$