Solve x(x-1)/2=1000

$$\frac{ x(x-1) }{ 2 } =1000$$

Solve for x

$$x = \frac{3 \sqrt{889} + 1}{2} \approx 45.224154548$$

$$x=\frac{1-3\sqrt{889}}{2}\approx -44.224154548$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

$$x\left(x-1\right)=1000\times 2$$

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

$$x^{2}-x=1000\times 2$$

Multiply 1000 and 2 to get 2000.

$$x^{2}-x=2000$$

Subtract 2000 from both sides.

$$x^{2}-x-2000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1 for b, and -2000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-2000\right)}}{2}$$

Multiply -4 times -2000.

$$x=\frac{-\left(-1\right)±\sqrt{1+8000}}{2}$$

Add 1 to 8000.

$$x=\frac{-\left(-1\right)±\sqrt{8001}}{2}$$

Take the square root of 8001.

$$x=\frac{-\left(-1\right)±3\sqrt{889}}{2}$$

The opposite of -1 is 1.

$$x=\frac{1±3\sqrt{889}}{2}$$

Now solve the equation x=\frac{1±3\sqrt{889}}{2} when ± is plus. Add 1 to 3\sqrt{889}.

$$x=\frac{3\sqrt{889}+1}{2}$$

Now solve the equation x=\frac{1±3\sqrt{889}}{2} when ± is minus. Subtract 3\sqrt{889} from 1.

$$x=\frac{1-3\sqrt{889}}{2}$$

The equation is now solved.

$$x=\frac{3\sqrt{889}+1}{2} x=\frac{1-3\sqrt{889}}{2}$$

Steps for Completing the Square

Multiply both sides by 2.

$$x\left(x-1\right)=1000\times 2$$

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

$$x^{2}-x=1000\times 2$$

Multiply 1000 and 2 to get 2000.

$$x^{2}-x=2000$$

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.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=2000+\left(-\frac{1}{2}\right)^{2}$$

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

$$x^{2}-x+\frac{1}{4}=2000+\frac{1}{4}$$

Add 2000 to \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{8001}{4}$$

Factor x^{2}-x+\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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{8001}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{8001}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{3\sqrt{889}}{2} x-\frac{1}{2}=-\frac{3\sqrt{889}}{2}$$

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

$$x=\frac{3\sqrt{889}+1}{2} x=\frac{1-3\sqrt{889}}{2}$$