Solve (1400-400x)(x-2)=224

$$( 1400 - 400 x ) ( x - 2 ) = 224$$

Solve for x

$$x = \frac{27}{10} = 2\frac{7}{10} = 2.7$$

$$x = \frac{14}{5} = 2\frac{4}{5} = 2.8$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 1400-400x by x-2 and combine like terms.

$$2200x-2800-400x^{2}=224$$

Subtract 224 from both sides.

$$2200x-2800-400x^{2}-224=0$$

Subtract 224 from -2800 to get -3024.

$$2200x-3024-400x^{2}=0$$

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.

$$-400x^{2}+2200x-3024=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -400 for a, 2200 for b, and -3024 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2200±\sqrt{2200^{2}-4\left(-400\right)\left(-3024\right)}}{2\left(-400\right)}$$

Square 2200.

$$x=\frac{-2200±\sqrt{4840000-4\left(-400\right)\left(-3024\right)}}{2\left(-400\right)}$$

Multiply -4 times -400.

$$x=\frac{-2200±\sqrt{4840000+1600\left(-3024\right)}}{2\left(-400\right)}$$

Multiply 1600 times -3024.

$$x=\frac{-2200±\sqrt{4840000-4838400}}{2\left(-400\right)}$$

Add 4840000 to -4838400.

$$x=\frac{-2200±\sqrt{1600}}{2\left(-400\right)}$$

Take the square root of 1600.

$$x=\frac{-2200±40}{2\left(-400\right)}$$

Multiply 2 times -400.

$$x=\frac{-2200±40}{-800}$$

Now solve the equation x=\frac{-2200±40}{-800} when ± is plus. Add -2200 to 40.

$$x=-\frac{2160}{-800}$$

Reduce the fraction \frac{-2160}{-800} to lowest terms by extracting and canceling out 80.

$$x=\frac{27}{10}$$

Now solve the equation x=\frac{-2200±40}{-800} when ± is minus. Subtract 40 from -2200.

$$x=-\frac{2240}{-800}$$

Reduce the fraction \frac{-2240}{-800} to lowest terms by extracting and canceling out 160.

$$x=\frac{14}{5}$$

The equation is now solved.

$$x=\frac{27}{10} x=\frac{14}{5}$$

Steps for Completing the Square

Use the distributive property to multiply 1400-400x by x-2 and combine like terms.

$$2200x-2800-400x^{2}=224$$

Add 2800 to both sides.

$$2200x-400x^{2}=224+2800$$

Add 224 and 2800 to get 3024.

$$2200x-400x^{2}=3024$$

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.

$$-400x^{2}+2200x=3024$$

Divide both sides by -400.

$$\frac{-400x^{2}+2200x}{-400}=\frac{3024}{-400}$$

Dividing by -400 undoes the multiplication by -400.

$$x^{2}+\frac{2200}{-400}x=\frac{3024}{-400}$$

Reduce the fraction \frac{2200}{-400} to lowest terms by extracting and canceling out 200.

$$x^{2}-\frac{11}{2}x=\frac{3024}{-400}$$

Reduce the fraction \frac{3024}{-400} to lowest terms by extracting and canceling out 16.

$$x^{2}-\frac{11}{2}x=-\frac{189}{25}$$

Divide -\frac{11}{2}, the coefficient of the x term, by 2 to get -\frac{11}{4}. Then add the square of -\frac{11}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{11}{2}x+\left(-\frac{11}{4}\right)^{2}=-\frac{189}{25}+\left(-\frac{11}{4}\right)^{2}$$

Square -\frac{11}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{11}{2}x+\frac{121}{16}=-\frac{189}{25}+\frac{121}{16}$$

Add -\frac{189}{25} to \frac{121}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{11}{2}x+\frac{121}{16}=\frac{1}{400}$$

Factor x^{2}-\frac{11}{2}x+\frac{121}{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}.

$$\left(x-\frac{11}{4}\right)^{2}=\frac{1}{400}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{4}\right)^{2}}=\sqrt{\frac{1}{400}}$$

Simplify.

$$x-\frac{11}{4}=\frac{1}{20} x-\frac{11}{4}=-\frac{1}{20}$$

Add \frac{11}{4} to both sides of the equation.

$$x=\frac{14}{5} x=\frac{27}{10}$$