Solve 0=x^2-100x+560000

$$0 = x ^ { 2 } - 100 x + 560000$$

Solve for x (complex solution)

$$x=50+50\sqrt{223}i\approx 50+746.659226153i$$

$$x=-50\sqrt{223}i+50\approx 50-746.659226153i$$

Steps Using the Quadratic Formula

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

$$x^{2}-100x+560000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -100 for b, and 560000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-100\right)±\sqrt{\left(-100\right)^{2}-4\times 560000}}{2}$$

Square -100.

$$x=\frac{-\left(-100\right)±\sqrt{10000-4\times 560000}}{2}$$

Multiply -4 times 560000.

$$x=\frac{-\left(-100\right)±\sqrt{10000-2240000}}{2}$$

Add 10000 to -2240000.

$$x=\frac{-\left(-100\right)±\sqrt{-2230000}}{2}$$

Take the square root of -2230000.

$$x=\frac{-\left(-100\right)±100\sqrt{223}i}{2}$$

The opposite of -100 is 100.

$$x=\frac{100±100\sqrt{223}i}{2}$$

Now solve the equation x=\frac{100±100\sqrt{223}i}{2} when ± is plus. Add 100 to 100i\sqrt{223}.

$$x=\frac{100+100\sqrt{223}i}{2}$$

Divide 100+100i\sqrt{223} by 2.

$$x=50+50\sqrt{223}i$$

Now solve the equation x=\frac{100±100\sqrt{223}i}{2} when ± is minus. Subtract 100i\sqrt{223} from 100.

$$x=\frac{-100\sqrt{223}i+100}{2}$$

Divide 100-100i\sqrt{223} by 2.

$$x=-50\sqrt{223}i+50$$

The equation is now solved.

$$x=50+50\sqrt{223}i x=-50\sqrt{223}i+50$$

Steps for Completing the Square

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

$$x^{2}-100x+560000=0$$

Subtract 560000 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-100x=-560000$$

Divide -100, the coefficient of the x term, by 2 to get -50. Then add the square of -50 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-100x+\left(-50\right)^{2}=-560000+\left(-50\right)^{2}$$

Square -50.

$$x^{2}-100x+2500=-560000+2500$$

Add -560000 to 2500.

$$x^{2}-100x+2500=-557500$$

Factor x^{2}-100x+2500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-50\right)^{2}=-557500$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-50\right)^{2}}=\sqrt{-557500}$$

Simplify.

$$x-50=50\sqrt{223}i x-50=-50\sqrt{223}i$$

Add 50 to both sides of the equation.

$$x=50+50\sqrt{223}i x=-50\sqrt{223}i+50$$