Solve n(n+1)-2048=0

$$n ( n + 1 ) - 2048 = 0$$

Solve for n

$$n = \frac{\sqrt{8193} - 1}{2} \approx 44.757596048$$

$$n=\frac{-\sqrt{8193}-1}{2}\approx -45.757596048$$

Steps Using the Quadratic Formula

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

$$n^{2}+n-2048=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 1 for b, and -2048 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-1±\sqrt{1^{2}-4\left(-2048\right)}}{2}$$

Square 1.

$$n=\frac{-1±\sqrt{1-4\left(-2048\right)}}{2}$$

Multiply -4 times -2048.

$$n=\frac{-1±\sqrt{1+8192}}{2}$$

Add 1 to 8192.

$$n=\frac{-1±\sqrt{8193}}{2}$$

Now solve the equation n=\frac{-1±\sqrt{8193}}{2} when ± is plus. Add -1 to \sqrt{8193}.

$$n=\frac{\sqrt{8193}-1}{2}$$

Now solve the equation n=\frac{-1±\sqrt{8193}}{2} when ± is minus. Subtract \sqrt{8193} from -1.

$$n=\frac{-\sqrt{8193}-1}{2}$$

The equation is now solved.

$$n=\frac{\sqrt{8193}-1}{2} n=\frac{-\sqrt{8193}-1}{2}$$

Steps for Completing the Square

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

$$n^{2}+n-2048=0$$

Add 2048 to both sides. Anything plus zero gives itself.

$$n^{2}+n=2048$$

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.

$$n^{2}+n+\left(\frac{1}{2}\right)^{2}=2048+\left(\frac{1}{2}\right)^{2}$$

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

$$n^{2}+n+\frac{1}{4}=2048+\frac{1}{4}$$

Add 2048 to \frac{1}{4}.

$$n^{2}+n+\frac{1}{4}=\frac{8193}{4}$$

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}.

$$\left(n+\frac{1}{2}\right)^{2}=\frac{8193}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+\frac{1}{2}\right)^{2}}=\sqrt{\frac{8193}{4}}$$

Simplify.

$$n+\frac{1}{2}=\frac{\sqrt{8193}}{2} n+\frac{1}{2}=-\frac{\sqrt{8193}}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$n=\frac{\sqrt{8193}-1}{2} n=\frac{-\sqrt{8193}-1}{2}$$