Solve (120+10x)(50-2x)=12

$$( 120 + 10 x ) ( 50 - 2 x ) = 12$$

Solve for x

$$x=\frac{\sqrt{34165}}{10}+\frac{13}{2}\approx 24.98377667$$

$$x=-\frac{\sqrt{34165}}{10}+\frac{13}{2}\approx -11.98377667$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 120+10x by 50-2x and combine like terms.

$$6000+260x-20x^{2}=12$$

Subtract 12 from both sides.

$$6000+260x-20x^{2}-12=0$$

Subtract 12 from 6000 to get 5988.

$$5988+260x-20x^{2}=0$$

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.

$$-20x^{2}+260x+5988=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -20 for a, 260 for b, and 5988 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-260±\sqrt{260^{2}-4\left(-20\right)\times 5988}}{2\left(-20\right)}$$

Square 260.

$$x=\frac{-260±\sqrt{67600-4\left(-20\right)\times 5988}}{2\left(-20\right)}$$

Multiply -4 times -20.

$$x=\frac{-260±\sqrt{67600+80\times 5988}}{2\left(-20\right)}$$

Multiply 80 times 5988.

$$x=\frac{-260±\sqrt{67600+479040}}{2\left(-20\right)}$$

Add 67600 to 479040.

$$x=\frac{-260±\sqrt{546640}}{2\left(-20\right)}$$

Take the square root of 546640.

$$x=\frac{-260±4\sqrt{34165}}{2\left(-20\right)}$$

Multiply 2 times -20.

$$x=\frac{-260±4\sqrt{34165}}{-40}$$

Now solve the equation x=\frac{-260±4\sqrt{34165}}{-40} when ± is plus. Add -260 to 4\sqrt{34165}.

$$x=\frac{4\sqrt{34165}-260}{-40}$$

Divide -260+4\sqrt{34165} by -40.

$$x=-\frac{\sqrt{34165}}{10}+\frac{13}{2}$$

Now solve the equation x=\frac{-260±4\sqrt{34165}}{-40} when ± is minus. Subtract 4\sqrt{34165} from -260.

$$x=\frac{-4\sqrt{34165}-260}{-40}$$

Divide -260-4\sqrt{34165} by -40.

$$x=\frac{\sqrt{34165}}{10}+\frac{13}{2}$$

The equation is now solved.

$$x=-\frac{\sqrt{34165}}{10}+\frac{13}{2} x=\frac{\sqrt{34165}}{10}+\frac{13}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 120+10x by 50-2x and combine like terms.

$$6000+260x-20x^{2}=12$$

Subtract 6000 from both sides.

$$260x-20x^{2}=12-6000$$

Subtract 6000 from 12 to get -5988.

$$260x-20x^{2}=-5988$$

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.

$$-20x^{2}+260x=-5988$$

Divide both sides by -20.

$$\frac{-20x^{2}+260x}{-20}=-\frac{5988}{-20}$$

Dividing by -20 undoes the multiplication by -20.

$$x^{2}+\frac{260}{-20}x=-\frac{5988}{-20}$$

Divide 260 by -20.

$$x^{2}-13x=-\frac{5988}{-20}$$

Reduce the fraction \frac{-5988}{-20} to lowest terms by extracting and canceling out 4.

$$x^{2}-13x=\frac{1497}{5}$$

Divide -13, the coefficient of the x term, by 2 to get -\frac{13}{2}. Then add the square of -\frac{13}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-13x+\left(-\frac{13}{2}\right)^{2}=\frac{1497}{5}+\left(-\frac{13}{2}\right)^{2}$$

Square -\frac{13}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-13x+\frac{169}{4}=\frac{1497}{5}+\frac{169}{4}$$

Add \frac{1497}{5} to \frac{169}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-13x+\frac{169}{4}=\frac{6833}{20}$$

Factor x^{2}-13x+\frac{169}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{13}{2}\right)^{2}=\frac{6833}{20}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{2}\right)^{2}}=\sqrt{\frac{6833}{20}}$$

Simplify.

$$x-\frac{13}{2}=\frac{\sqrt{34165}}{10} x-\frac{13}{2}=-\frac{\sqrt{34165}}{10}$$

Add \frac{13}{2} to both sides of the equation.

$$x=\frac{\sqrt{34165}}{10}+\frac{13}{2} x=-\frac{\sqrt{34165}}{10}+\frac{13}{2}$$