Solve (1-x)(200+400x)=200

$$(1-x)(200+400x)=200$$

Solve for x

$$x=\frac{1}{2}=0.5$$

$$x=0$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 1-x by 200+400x and combine like terms.

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

Subtract 200 from both sides.

$$200+200x-400x^{2}-200=0$$

Subtract 200 from 200 to get 0.

$$200x-400x^{2}=0$$

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.

$$-400x^{2}+200x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -400 for a, 200 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-200±\sqrt{200^{2}}}{2\left(-400\right)}$$

Take the square root of 200^{2}.

$$x=\frac{-200±200}{2\left(-400\right)}$$

Multiply 2 times -400.

$$x=\frac{-200±200}{-800}$$

Now solve the equation x=\frac{-200±200}{-800} when ± is plus. Add -200 to 200.

$$x=\frac{0}{-800}$$

Divide 0 by -800.

$$x=0$$

Now solve the equation x=\frac{-200±200}{-800} when ± is minus. Subtract 200 from -200.

$$x=-\frac{400}{-800}$$

Reduce the fraction \frac{-400}{-800} to lowest terms by extracting and canceling out 400.

$$x=\frac{1}{2}$$

The equation is now solved.

$$x=0 x=\frac{1}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 1-x by 200+400x and combine like terms.

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

Subtract 200 from both sides.

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

Subtract 200 from 200 to get 0.

$$200x-400x^{2}=0$$

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.

$$-400x^{2}+200x=0$$

Divide both sides by -400.

$$\frac{-400x^{2}+200x}{-400}=\frac{0}{-400}$$

Dividing by -400 undoes the multiplication by -400.

$$x^{2}+\frac{200}{-400}x=\frac{0}{-400}$$

Reduce the fraction \frac{200}{-400} to lowest terms by extracting and canceling out 200.

$$x^{2}-\frac{1}{2}x=\frac{0}{-400}$$

Divide 0 by -400.

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

Divide -\frac{1}{2}, the coefficient of the x term, by 2 to get -\frac{1}{4}. Then add the square of -\frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=\left(-\frac{1}{4}\right)^{2}$$

Square -\frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{1}{16}$$

Factor x^{2}-\frac{1}{2}x+\frac{1}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{4}\right)^{2}=\frac{1}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{\frac{1}{16}}$$

Simplify.

$$x-\frac{1}{4}=\frac{1}{4} x-\frac{1}{4}=-\frac{1}{4}$$

Add \frac{1}{4} to both sides of the equation.

$$x=\frac{1}{2} x=0$$