Solve -x^2+500x+1=0

$$- { x }^{ 2 } +500x+1=0$$

Solve for x

$$x=\sqrt{62501}+250\approx 500.001999992$$

$$x=250-\sqrt{62501}\approx -0.001999992$$

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.

$$-x^{2}+500x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 500 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-500±\sqrt{500^{2}-4\left(-1\right)}}{2\left(-1\right)}$$

Square 500.

$$x=\frac{-500±\sqrt{250000-4\left(-1\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-500±\sqrt{250000+4}}{2\left(-1\right)}$$

Add 250000 to 4.

$$x=\frac{-500±\sqrt{250004}}{2\left(-1\right)}$$

Take the square root of 250004.

$$x=\frac{-500±2\sqrt{62501}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-500±2\sqrt{62501}}{-2}$$

Now solve the equation x=\frac{-500±2\sqrt{62501}}{-2} when ± is plus. Add -500 to 2\sqrt{62501}.

$$x=\frac{2\sqrt{62501}-500}{-2}$$

Divide -500+2\sqrt{62501} by -2.

$$x=250-\sqrt{62501}$$

Now solve the equation x=\frac{-500±2\sqrt{62501}}{-2} when ± is minus. Subtract 2\sqrt{62501} from -500.

$$x=\frac{-2\sqrt{62501}-500}{-2}$$

Divide -500-2\sqrt{62501} by -2.

$$x=\sqrt{62501}+250$$

The equation is now solved.

$$x=250-\sqrt{62501} x=\sqrt{62501}+250$$

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.

$$-x^{2}+500x+1=0$$

Subtract 1 from both sides of the equation.

$$-x^{2}+500x+1-1=-1$$

Subtracting 1 from itself leaves 0.

$$-x^{2}+500x=-1$$

Divide both sides by -1.

$$\frac{-x^{2}+500x}{-1}=-\frac{1}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{500}{-1}x=-\frac{1}{-1}$$

Divide 500 by -1.

$$x^{2}-500x=-\frac{1}{-1}$$

Divide -1 by -1.

$$x^{2}-500x=1$$

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.

$$x^{2}-500x+\left(-250\right)^{2}=1+\left(-250\right)^{2}$$

Square -250.

$$x^{2}-500x+62500=1+62500$$

Add 1 to 62500.

$$x^{2}-500x+62500=62501$$

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}.

$$\left(x-250\right)^{2}=62501$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-250\right)^{2}}=\sqrt{62501}$$

Simplify.

$$x-250=\sqrt{62501} x-250=-\sqrt{62501}$$

Add 250 to both sides of the equation.

$$x=\sqrt{62501}+250 x=250-\sqrt{62501}$$