Solve 6=-4x^2+2000x

$$6 = - 4 x ^ { 2 } + 2000 x$$

Solve for x

$$x=\frac{\sqrt{249994}}{2}+250\approx 499.996999982$$

$$x=-\frac{\sqrt{249994}}{2}+250\approx 0.003000018$$

Steps Using the Quadratic Formula

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

$$-4x^{2}+2000x=6$$

Subtract 6 from both sides.

$$-4x^{2}+2000x-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 2000 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2000±\sqrt{2000^{2}-4\left(-4\right)\left(-6\right)}}{2\left(-4\right)}$$

Square 2000.

$$x=\frac{-2000±\sqrt{4000000-4\left(-4\right)\left(-6\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-2000±\sqrt{4000000+16\left(-6\right)}}{2\left(-4\right)}$$

Multiply 16 times -6.

$$x=\frac{-2000±\sqrt{4000000-96}}{2\left(-4\right)}$$

Add 4000000 to -96.

$$x=\frac{-2000±\sqrt{3999904}}{2\left(-4\right)}$$

Take the square root of 3999904.

$$x=\frac{-2000±4\sqrt{249994}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-2000±4\sqrt{249994}}{-8}$$

Now solve the equation x=\frac{-2000±4\sqrt{249994}}{-8} when ± is plus. Add -2000 to 4\sqrt{249994}.

$$x=\frac{4\sqrt{249994}-2000}{-8}$$

Divide -2000+4\sqrt{249994} by -8.

$$x=-\frac{\sqrt{249994}}{2}+250$$

Now solve the equation x=\frac{-2000±4\sqrt{249994}}{-8} when ± is minus. Subtract 4\sqrt{249994} from -2000.

$$x=\frac{-4\sqrt{249994}-2000}{-8}$$

Divide -2000-4\sqrt{249994} by -8.

$$x=\frac{\sqrt{249994}}{2}+250$$

The equation is now solved.

$$x=-\frac{\sqrt{249994}}{2}+250 x=\frac{\sqrt{249994}}{2}+250$$

Steps for Completing the Square

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

$$-4x^{2}+2000x=6$$

Divide both sides by -4.

$$\frac{-4x^{2}+2000x}{-4}=\frac{6}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{2000}{-4}x=\frac{6}{-4}$$

Divide 2000 by -4.

$$x^{2}-500x=\frac{6}{-4}$$

Reduce the fraction \frac{6}{-4} to lowest terms by extracting and canceling out 2.

$$x^{2}-500x=-\frac{3}{2}$$

Divide -500, the coefficient of the x term, by 2 to get -250. Then add the square of -250 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-500x+\left(-250\right)^{2}=-\frac{3}{2}+\left(-250\right)^{2}$$

Square -250.

$$x^{2}-500x+62500=-\frac{3}{2}+62500$$

Add -\frac{3}{2} to 62500.

$$x^{2}-500x+62500=\frac{124997}{2}$$

Factor x^{2}-500x+62500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-250\right)^{2}=\frac{124997}{2}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-250\right)^{2}}=\sqrt{\frac{124997}{2}}$$

Simplify.

$$x-250=\frac{\sqrt{249994}}{2} x-250=-\frac{\sqrt{249994}}{2}$$

Add 250 to both sides of the equation.

$$x=\frac{\sqrt{249994}}{2}+250 x=-\frac{\sqrt{249994}}{2}+250$$