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

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

Solve for x

$$x=100$$

$$x=0$$

Steps Using Factoring

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

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

Subtract 400 from both sides.

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

Subtract 400 from 400 to get 0.

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

Factor out x.

$$x\left(50-0.5x\right)=0$$

To find equation solutions, solve x=0 and 50-\frac{x}{2}=0.

$$x=0 x=100$$

Steps Using the Quadratic Formula

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

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

Subtract 400 from both sides.

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

Subtract 400 from 400 to get 0.

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

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.

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

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

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

Take the square root of 50^{2}.

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

Multiply 2 times -0.5.

$$x=\frac{-50±50}{-1}$$

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

$$x=\frac{0}{-1}$$

Divide 0 by -1.

$$x=0$$

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

$$x=-\frac{100}{-1}$$

Divide -100 by -1.

$$x=100$$

The equation is now solved.

$$x=0 x=100$$

Steps for Completing the Square

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

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

Subtract 400 from both sides.

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

Subtract 400 from 400 to get 0.

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

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.

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

Multiply both sides by -2.

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

Dividing by -0.5 undoes the multiplication by -0.5.

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

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

$$x^{2}-100x=\frac{0}{-0.5}$$

Divide 0 by -0.5 by multiplying 0 by the reciprocal of -0.5.

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

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.

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

Square -50.

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

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}.

$$\left(x-50\right)^{2}=2500$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-50\right)^{2}}=\sqrt{2500}$$

Simplify.

$$x-50=50 x-50=-50$$

Add 50 to both sides of the equation.

$$x=100 x=0$$