Solve x^2-20x+570=0

$${ x }^{ 2 } -20x+570=0$$

Solve for x (complex solution)

$$x=10+\sqrt{470}i\approx 10+21.679483389i$$

$$x=-\sqrt{470}i+10\approx 10-21.679483389i$$

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.

$$x^{2}-20x+570=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -20 for b, and 570 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 570}}{2}$$

Square -20.

$$x=\frac{-\left(-20\right)±\sqrt{400-4\times 570}}{2}$$

Multiply -4 times 570.

$$x=\frac{-\left(-20\right)±\sqrt{400-2280}}{2}$$

Add 400 to -2280.

$$x=\frac{-\left(-20\right)±\sqrt{-1880}}{2}$$

Take the square root of -1880.

$$x=\frac{-\left(-20\right)±2\sqrt{470}i}{2}$$

The opposite of -20 is 20.

$$x=\frac{20±2\sqrt{470}i}{2}$$

Now solve the equation x=\frac{20±2\sqrt{470}i}{2} when ± is plus. Add 20 to 2i\sqrt{470}.

$$x=\frac{20+2\sqrt{470}i}{2}$$

Divide 20+2i\sqrt{470} by 2.

$$x=10+\sqrt{470}i$$

Now solve the equation x=\frac{20±2\sqrt{470}i}{2} when ± is minus. Subtract 2i\sqrt{470} from 20.

$$x=\frac{-2\sqrt{470}i+20}{2}$$

Divide 20-2i\sqrt{470} by 2.

$$x=-\sqrt{470}i+10$$

The equation is now solved.

$$x=10+\sqrt{470}i x=-\sqrt{470}i+10$$

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.

$$x^{2}-20x+570=0$$

Subtract 570 from both sides of the equation.

$$x^{2}-20x+570-570=-570$$

Subtracting 570 from itself leaves 0.

$$x^{2}-20x=-570$$

Divide -20, the coefficient of the x term, by 2 to get -10. Then add the square of -10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-20x+\left(-10\right)^{2}=-570+\left(-10\right)^{2}$$

Square -10.

$$x^{2}-20x+100=-570+100$$

Add -570 to 100.

$$x^{2}-20x+100=-470$$

Factor x^{2}-20x+100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-10\right)^{2}=-470$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-10\right)^{2}}=\sqrt{-470}$$

Simplify.

$$x-10=\sqrt{470}i x-10=-\sqrt{470}i$$

Add 10 to both sides of the equation.

$$x=10+\sqrt{470}i x=-\sqrt{470}i+10$$