Solve left(1-xright)(200+400x)=224

$$\left( 1-x \right) (200+400x)=224$$

Solve for x

$$x=\frac{3}{10}=0.3$$

$$x=\frac{1}{5}=0.2$$

Steps Using the Quadratic Formula

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

$$200+200x-400x^{2}=224$$

Subtract 224 from both sides.

$$200+200x-400x^{2}-224=0$$

Subtract 224 from 200 to get -24.

$$-24+200x-400x^{2}=0$$

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.

$$-400x^{2}+200x-24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -400 for a, 200 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-200±\sqrt{200^{2}-4\left(-400\right)\left(-24\right)}}{2\left(-400\right)}$$

Square 200.

$$x=\frac{-200±\sqrt{40000-4\left(-400\right)\left(-24\right)}}{2\left(-400\right)}$$

Multiply -4 times -400.

$$x=\frac{-200±\sqrt{40000+1600\left(-24\right)}}{2\left(-400\right)}$$

Multiply 1600 times -24.

$$x=\frac{-200±\sqrt{40000-38400}}{2\left(-400\right)}$$

Add 40000 to -38400.

$$x=\frac{-200±\sqrt{1600}}{2\left(-400\right)}$$

Take the square root of 1600.

$$x=\frac{-200±40}{2\left(-400\right)}$$

Multiply 2 times -400.

$$x=\frac{-200±40}{-800}$$

Now solve the equation x=\frac{-200±40}{-800} when ± is plus. Add -200 to 40.

$$x=-\frac{160}{-800}$$

Reduce the fraction \frac{-160}{-800} to lowest terms by extracting and canceling out 160.

$$x=\frac{1}{5}$$

Now solve the equation x=\frac{-200±40}{-800} when ± is minus. Subtract 40 from -200.

$$x=-\frac{240}{-800}$$

Reduce the fraction \frac{-240}{-800} to lowest terms by extracting and canceling out 80.

$$x=\frac{3}{10}$$

The equation is now solved.

$$x=\frac{1}{5} x=\frac{3}{10}$$

Steps for Completing the Square

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

$$200+200x-400x^{2}=224$$

Subtract 200 from both sides.

$$200x-400x^{2}=224-200$$

Subtract 200 from 224 to get 24.

$$200x-400x^{2}=24$$

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.

$$-400x^{2}+200x=24$$

Divide both sides by -400.

$$\frac{-400x^{2}+200x}{-400}=\frac{24}{-400}$$

Dividing by -400 undoes the multiplication by -400.

$$x^{2}+\frac{200}{-400}x=\frac{24}{-400}$$

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

$$x^{2}-\frac{1}{2}x=\frac{24}{-400}$$

Reduce the fraction \frac{24}{-400} to lowest terms by extracting and canceling out 8.

$$x^{2}-\frac{1}{2}x=-\frac{3}{50}$$

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.

$$x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=-\frac{3}{50}+\left(-\frac{1}{4}\right)^{2}$$

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

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=-\frac{3}{50}+\frac{1}{16}$$

Add -\frac{3}{50} to \frac{1}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{1}{400}$$

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}.

$$\left(x-\frac{1}{4}\right)^{2}=\frac{1}{400}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{\frac{1}{400}}$$

Simplify.

$$x-\frac{1}{4}=\frac{1}{20} x-\frac{1}{4}=-\frac{1}{20}$$

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

$$x=\frac{3}{10} x=\frac{1}{5}$$