Solve 2000000=x/2(2000000+(x-1)2000)

$$2000000= \frac{ x }{ 2 } (2000000+(x-1)2000)$$

Solve for x

$$x = \frac{\sqrt{1006001} - 999}{2} \approx 1.998005978$$

$$x=\frac{-\sqrt{1006001}-999}{2}\approx -1000.998005978$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$4000000=x\left(2000000+\left(x-1\right)\times 2000\right)$$

Use the distributive property to multiply x-1 by 2000.

$$4000000=x\left(2000000+2000x-2000\right)$$

Subtract 2000 from 2000000 to get 1998000.

$$4000000=x\left(1998000+2000x\right)$$

Use the distributive property to multiply x by 1998000+2000x.

$$4000000=1998000x+2000x^{2}$$

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

$$1998000x+2000x^{2}=4000000$$

Subtract 4000000 from both sides.

$$1998000x+2000x^{2}-4000000=0$$

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.

$$2000x^{2}+1998000x-4000000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2000 for a, 1998000 for b, and -4000000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1998000±\sqrt{1998000^{2}-4\times 2000\left(-4000000\right)}}{2\times 2000}$$

Square 1998000.

$$x=\frac{-1998000±\sqrt{3992004000000-4\times 2000\left(-4000000\right)}}{2\times 2000}$$

Multiply -4 times 2000.

$$x=\frac{-1998000±\sqrt{3992004000000-8000\left(-4000000\right)}}{2\times 2000}$$

Multiply -8000 times -4000000.

$$x=\frac{-1998000±\sqrt{3992004000000+32000000000}}{2\times 2000}$$

Add 3992004000000 to 32000000000.

$$x=\frac{-1998000±\sqrt{4024004000000}}{2\times 2000}$$

Take the square root of 4024004000000.

$$x=\frac{-1998000±2000\sqrt{1006001}}{2\times 2000}$$

Multiply 2 times 2000.

$$x=\frac{-1998000±2000\sqrt{1006001}}{4000}$$

Now solve the equation x=\frac{-1998000±2000\sqrt{1006001}}{4000} when ± is plus. Add -1998000 to 2000\sqrt{1006001}.

$$x=\frac{2000\sqrt{1006001}-1998000}{4000}$$

Divide -1998000+2000\sqrt{1006001} by 4000.

$$x=\frac{\sqrt{1006001}-999}{2}$$

Now solve the equation x=\frac{-1998000±2000\sqrt{1006001}}{4000} when ± is minus. Subtract 2000\sqrt{1006001} from -1998000.

$$x=\frac{-2000\sqrt{1006001}-1998000}{4000}$$

Divide -1998000-2000\sqrt{1006001} by 4000.

$$x=\frac{-\sqrt{1006001}-999}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{1006001}-999}{2} x=\frac{-\sqrt{1006001}-999}{2}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$4000000=x\left(2000000+\left(x-1\right)\times 2000\right)$$

Use the distributive property to multiply x-1 by 2000.

$$4000000=x\left(2000000+2000x-2000\right)$$

Subtract 2000 from 2000000 to get 1998000.

$$4000000=x\left(1998000+2000x\right)$$

Use the distributive property to multiply x by 1998000+2000x.

$$4000000=1998000x+2000x^{2}$$

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

$$1998000x+2000x^{2}=4000000$$

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.

$$2000x^{2}+1998000x=4000000$$

Divide both sides by 2000.

$$\frac{2000x^{2}+1998000x}{2000}=\frac{4000000}{2000}$$

Dividing by 2000 undoes the multiplication by 2000.

$$x^{2}+\frac{1998000}{2000}x=\frac{4000000}{2000}$$

Divide 1998000 by 2000.

$$x^{2}+999x=\frac{4000000}{2000}$$

Divide 4000000 by 2000.

$$x^{2}+999x=2000$$

Divide 999, the coefficient of the x term, by 2 to get \frac{999}{2}. Then add the square of \frac{999}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+999x+\left(\frac{999}{2}\right)^{2}=2000+\left(\frac{999}{2}\right)^{2}$$

Square \frac{999}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+999x+\frac{998001}{4}=2000+\frac{998001}{4}$$

Add 2000 to \frac{998001}{4}.

$$x^{2}+999x+\frac{998001}{4}=\frac{1006001}{4}$$

Factor x^{2}+999x+\frac{998001}{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}.

$$\left(x+\frac{999}{2}\right)^{2}=\frac{1006001}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{999}{2}\right)^{2}}=\sqrt{\frac{1006001}{4}}$$

Simplify.

$$x+\frac{999}{2}=\frac{\sqrt{1006001}}{2} x+\frac{999}{2}=-\frac{\sqrt{1006001}}{2}$$

Subtract \frac{999}{2} from both sides of the equation.

$$x=\frac{\sqrt{1006001}-999}{2} x=\frac{-\sqrt{1006001}-999}{2}$$