Solve 100x-0.5x^2-50x+400=0

$$100x-0.5 { x }^{ 2 } -50x+400=0$$

Solve for x

$$x=10\sqrt{33}+50\approx 107.445626465$$

$$x=50-10\sqrt{33}\approx -7.445626465$$

Steps Using the Quadratic Formula

Combine 100x and -50x to get 50x.

$$50x-0.5x^{2}+400=0$$

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.

$$-0.5x^{2}+50x+400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.5 for a, 50 for b, and 400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-50±\sqrt{50^{2}-4\left(-0.5\right)\times 400}}{2\left(-0.5\right)}$$

Square 50.

$$x=\frac{-50±\sqrt{2500-4\left(-0.5\right)\times 400}}{2\left(-0.5\right)}$$

Multiply -4 times -0.5.

$$x=\frac{-50±\sqrt{2500+2\times 400}}{2\left(-0.5\right)}$$

Multiply 2 times 400.

$$x=\frac{-50±\sqrt{2500+800}}{2\left(-0.5\right)}$$

Add 2500 to 800.

$$x=\frac{-50±\sqrt{3300}}{2\left(-0.5\right)}$$

Take the square root of 3300.

$$x=\frac{-50±10\sqrt{33}}{2\left(-0.5\right)}$$

Multiply 2 times -0.5.

$$x=\frac{-50±10\sqrt{33}}{-1}$$

Now solve the equation x=\frac{-50±10\sqrt{33}}{-1} when ± is plus. Add -50 to 10\sqrt{33}.

$$x=\frac{10\sqrt{33}-50}{-1}$$

Divide -50+10\sqrt{33} by -1.

$$x=50-10\sqrt{33}$$

Now solve the equation x=\frac{-50±10\sqrt{33}}{-1} when ± is minus. Subtract 10\sqrt{33} from -50.

$$x=\frac{-10\sqrt{33}-50}{-1}$$

Divide -50-10\sqrt{33} by -1.

$$x=10\sqrt{33}+50$$

The equation is now solved.

$$x=50-10\sqrt{33} x=10\sqrt{33}+50$$

Steps for Completing the Square

Combine 100x and -50x to get 50x.

$$50x-0.5x^{2}+400=0$$

Subtract 400 from both sides. Anything subtracted from zero gives its negation.

$$50x-0.5x^{2}=-400$$

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.

$$-0.5x^{2}+50x=-400$$

Multiply both sides by -2.

$$\frac{-0.5x^{2}+50x}{-0.5}=-\frac{400}{-0.5}$$

Dividing by -0.5 undoes the multiplication by -0.5.

$$x^{2}+\frac{50}{-0.5}x=-\frac{400}{-0.5}$$

Divide 50 by -0.5 by multiplying 50 by the reciprocal of -0.5.

$$x^{2}-100x=-\frac{400}{-0.5}$$

Divide -400 by -0.5 by multiplying -400 by the reciprocal of -0.5.

$$x^{2}-100x=800$$

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.

$$x^{2}-100x+\left(-50\right)^{2}=800+\left(-50\right)^{2}$$

Square -50.

$$x^{2}-100x+2500=800+2500$$

Add 800 to 2500.

$$x^{2}-100x+2500=3300$$

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}.

$$\left(x-50\right)^{2}=3300$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-50\right)^{2}}=\sqrt{3300}$$

Simplify.

$$x-50=10\sqrt{33} x-50=-10\sqrt{33}$$

Add 50 to both sides of the equation.

$$x=10\sqrt{33}+50 x=50-10\sqrt{33}$$