Solve ex^2+2000x-2001=0

$$ex ^ { 2 } + 2000 x - 2001 = 0$$

Solve for x

$$x=\frac{\sqrt{2001e+1000000}-1000}{e}\approx 0.999143187$$

$$x=-\frac{\sqrt{2001e+1000000}+1000}{e}\approx -736.75802553$$

Steps Using the Quadratic Formula

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.

$$ex^{2}+2000x-2001=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute e for a, 2000 for b, and -2001 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2000±\sqrt{2000^{2}-4e\left(-2001\right)}}{2e}$$

Square 2000.

$$x=\frac{-2000±\sqrt{4000000-4e\left(-2001\right)}}{2e}$$

Multiply -4 times e.

$$x=\frac{-2000±\sqrt{4000000+\left(-4e\right)\left(-2001\right)}}{2e}$$

Multiply -4e times -2001.

$$x=\frac{-2000±\sqrt{4000000+8004e}}{2e}$$

Add 4000000 to 8004e.

$$x=\frac{-2000±\sqrt{8004e+4000000}}{2e}$$

Take the square root of 4000000+8004e.

$$x=\frac{-2000±2\sqrt{2001e+1000000}}{2e}$$

Now solve the equation x=\frac{-2000±2\sqrt{2001e+1000000}}{2e} when ± is plus. Add -2000 to 2\sqrt{1000000+2001e}.

$$x=\frac{2\sqrt{2001e+1000000}-2000}{2e}$$

Divide -2000+2\sqrt{1000000+2001e} by 2e.

$$x=\frac{\sqrt{2001e+1000000}-1000}{e}$$

Now solve the equation x=\frac{-2000±2\sqrt{2001e+1000000}}{2e} when ± is minus. Subtract 2\sqrt{1000000+2001e} from -2000.

$$x=\frac{-2\sqrt{2001e+1000000}-2000}{2e}$$

Divide -2000-2\sqrt{1000000+2001e} by 2e.

$$x=-\frac{\sqrt{2001e+1000000}+1000}{e}$$

The equation is now solved.

$$x=\frac{\sqrt{2001e+1000000}-1000}{e} x=-\frac{\sqrt{2001e+1000000}+1000}{e}$$

Steps for Completing the Square

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.

$$ex^{2}+2000x-2001=0$$

Add 2001 to both sides of the equation.

$$ex^{2}+2000x-2001-\left(-2001\right)=-\left(-2001\right)$$

Subtracting -2001 from itself leaves 0.

$$ex^{2}+2000x=-\left(-2001\right)$$

Subtract -2001 from 0.

$$ex^{2}+2000x=2001$$

Divide both sides by e.

$$\frac{ex^{2}+2000x}{e}=\frac{2001}{e}$$

Dividing by e undoes the multiplication by e.

$$x^{2}+\frac{2000}{e}x=\frac{2001}{e}$$

Divide \frac{2000}{e}, the coefficient of the x term, by 2 to get \frac{1000}{e}. Then add the square of \frac{1000}{e} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{2000}{e}x+\left(\frac{1000}{e}\right)^{2}=\frac{2001}{e}+\left(\frac{1000}{e}\right)^{2}$$

Square \frac{1000}{e}.

$$x^{2}+\frac{2000}{e}x+\frac{1000000}{e^{2}}=\frac{2001}{e}+\frac{1000000}{e^{2}}$$

Add \frac{2001}{e} to \frac{1000000}{e^{2}}.

$$x^{2}+\frac{2000}{e}x+\frac{1000000}{e^{2}}=\frac{2001e+1000000}{e^{2}}$$

Factor x^{2}+\frac{2000}{e}x+\frac{1000000}{e^{2}}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1000}{e}\right)^{2}=\frac{2001e+1000000}{e^{2}}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1000}{e}\right)^{2}}=\sqrt{\frac{2001e+1000000}{e^{2}}}$$

Simplify.

$$x+\frac{1000}{e}=\frac{\sqrt{2001e+1000000}}{e} x+\frac{1000}{e}=-\frac{\sqrt{2001e+1000000}}{e}$$

Subtract \frac{1000}{e} from both sides of the equation.

$$x=\frac{\sqrt{2001e+1000000}-1000}{e} x=-\frac{\sqrt{2001e+1000000}+1000}{e}$$