Solve -x^2+1000x+1=0

$$- { x }^{ 2 } +1000x+1=0$$

Solve for x

$$x=53\sqrt{89}+500\approx 1000.000999999$$

$$x=500-53\sqrt{89}\approx -0.000999999$$

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.

$$-x^{2}+1000x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 1000 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1000±\sqrt{1000^{2}-4\left(-1\right)}}{2\left(-1\right)}$$

Square 1000.

$$x=\frac{-1000±\sqrt{1000000-4\left(-1\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-1000±\sqrt{1000000+4}}{2\left(-1\right)}$$

Add 1000000 to 4.

$$x=\frac{-1000±\sqrt{1000004}}{2\left(-1\right)}$$

Take the square root of 1000004.

$$x=\frac{-1000±106\sqrt{89}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-1000±106\sqrt{89}}{-2}$$

Now solve the equation x=\frac{-1000±106\sqrt{89}}{-2} when ± is plus. Add -1000 to 106\sqrt{89}.

$$x=\frac{106\sqrt{89}-1000}{-2}$$

Divide -1000+106\sqrt{89} by -2.

$$x=500-53\sqrt{89}$$

Now solve the equation x=\frac{-1000±106\sqrt{89}}{-2} when ± is minus. Subtract 106\sqrt{89} from -1000.

$$x=\frac{-106\sqrt{89}-1000}{-2}$$

Divide -1000-106\sqrt{89} by -2.

$$x=53\sqrt{89}+500$$

The equation is now solved.

$$x=500-53\sqrt{89} x=53\sqrt{89}+500$$

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.

$$-x^{2}+1000x+1=0$$

Subtract 1 from both sides of the equation.

$$-x^{2}+1000x+1-1=-1$$

Subtracting 1 from itself leaves 0.

$$-x^{2}+1000x=-1$$

Divide both sides by -1.

$$\frac{-x^{2}+1000x}{-1}=-\frac{1}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{1000}{-1}x=-\frac{1}{-1}$$

Divide 1000 by -1.

$$x^{2}-1000x=-\frac{1}{-1}$$

Divide -1 by -1.

$$x^{2}-1000x=1$$

Divide -1000, the coefficient of the x term, by 2 to get -500. Then add the square of -500 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-1000x+\left(-500\right)^{2}=1+\left(-500\right)^{2}$$

Square -500.

$$x^{2}-1000x+250000=1+250000$$

Add 1 to 250000.

$$x^{2}-1000x+250000=250001$$

Factor x^{2}-1000x+250000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-500\right)^{2}=250001$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-500\right)^{2}}=\sqrt{250001}$$

Simplify.

$$x-500=53\sqrt{89} x-500=-53\sqrt{89}$$

Add 500 to both sides of the equation.

$$x=53\sqrt{89}+500 x=500-53\sqrt{89}$$