Solve x(-x/2+100)-50x+400=400

$$x(- \frac{ x }{ 2 } +100)-50x+400=400$$

Solve for x

$$x=100$$

$$x=0$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

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

Use the distributive property to multiply 2x by -\frac{x}{2}+100.

$$2x\left(-\frac{x}{2}\right)+200x-100x+800=800$$

Express 2\left(-\frac{x}{2}\right) as a single fraction.

$$\frac{-2x}{2}x+200x-100x+800=800$$

Cancel out 2 and 2.

$$-xx+200x-100x+800=800$$

Combine 200x and -100x to get 100x.

$$-xx+100x+800=800$$

Multiply x and x to get x^{2}.

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

Subtract 800 from both sides.

$$-x^{2}+100x+800-800=0$$

Subtract 800 from 800 to get 0.

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

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

$$x=\frac{-100±\sqrt{100^{2}}}{2\left(-1\right)}$$

Take the square root of 100^{2}.

$$x=\frac{-100±100}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-100±100}{-2}$$

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

$$x=\frac{0}{-2}$$

Divide 0 by -2.

$$x=0$$

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

$$x=-\frac{200}{-2}$$

Divide -200 by -2.

$$x=100$$

The equation is now solved.

$$x=0 x=100$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

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

Use the distributive property to multiply 2x by -\frac{x}{2}+100.

$$2x\left(-\frac{x}{2}\right)+200x-100x+800=800$$

Express 2\left(-\frac{x}{2}\right) as a single fraction.

$$\frac{-2x}{2}x+200x-100x+800=800$$

Cancel out 2 and 2.

$$-xx+200x-100x+800=800$$

Combine 200x and -100x to get 100x.

$$-xx+100x+800=800$$

Multiply x and x to get x^{2}.

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

Subtract 800 from both sides.

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

Subtract 800 from 800 to get 0.

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

Divide both sides by -1.

$$\frac{-x^{2}+100x}{-1}=\frac{0}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{100}{-1}x=\frac{0}{-1}$$

Divide 100 by -1.

$$x^{2}-100x=\frac{0}{-1}$$

Divide 0 by -1.

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

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.

$$x=100 x=0$$