Solve x^2-40x+200=0

$${ x }^{ 2 } -40x+200=0$$

Solve for x

$$x=10\sqrt{2}+20\approx 34.142135624$$

$$x=20-10\sqrt{2}\approx 5.857864376$$

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.

$$x^{2}-40x+200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -40 for b, and 200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-40\right)±\sqrt{\left(-40\right)^{2}-4\times 200}}{2}$$

Square -40.

$$x=\frac{-\left(-40\right)±\sqrt{1600-4\times 200}}{2}$$

Multiply -4 times 200.

$$x=\frac{-\left(-40\right)±\sqrt{1600-800}}{2}$$

Add 1600 to -800.

$$x=\frac{-\left(-40\right)±\sqrt{800}}{2}$$

Take the square root of 800.

$$x=\frac{-\left(-40\right)±20\sqrt{2}}{2}$$

The opposite of -40 is 40.

$$x=\frac{40±20\sqrt{2}}{2}$$

Now solve the equation x=\frac{40±20\sqrt{2}}{2} when ± is plus. Add 40 to 20\sqrt{2}.

$$x=\frac{20\sqrt{2}+40}{2}$$

Divide 40+20\sqrt{2} by 2.

$$x=10\sqrt{2}+20$$

Now solve the equation x=\frac{40±20\sqrt{2}}{2} when ± is minus. Subtract 20\sqrt{2} from 40.

$$x=\frac{40-20\sqrt{2}}{2}$$

Divide 40-20\sqrt{2} by 2.

$$x=20-10\sqrt{2}$$

The equation is now solved.

$$x=10\sqrt{2}+20 x=20-10\sqrt{2}$$

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.

$$x^{2}-40x+200=0$$

Subtract 200 from both sides of the equation.

$$x^{2}-40x+200-200=-200$$

Subtracting 200 from itself leaves 0.

$$x^{2}-40x=-200$$

Divide -40, the coefficient of the x term, by 2 to get -20. Then add the square of -20 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-40x+\left(-20\right)^{2}=-200+\left(-20\right)^{2}$$

Square -20.

$$x^{2}-40x+400=-200+400$$

Add -200 to 400.

$$x^{2}-40x+400=200$$

Factor x^{2}-40x+400. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-20\right)^{2}=200$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-20\right)^{2}}=\sqrt{200}$$

Simplify.

$$x-20=10\sqrt{2} x-20=-10\sqrt{2}$$

Add 20 to both sides of the equation.

$$x=10\sqrt{2}+20 x=20-10\sqrt{2}$$