Solve (-2/5x+20)(12+x)=384

$$( - \frac { 2 } { 5 } x + 20 ) ( 12 + x ) = 384$$

Solve for x

$$x=18$$

$$x=20$$

Steps Using the Quadratic Formula

Use the distributive property to multiply -\frac{2}{5}x+20 by 12+x and combine like terms.

$$\frac{76}{5}x-\frac{2}{5}x^{2}+240=384$$

Subtract 384 from both sides.

$$\frac{76}{5}x-\frac{2}{5}x^{2}+240-384=0$$

Subtract 384 from 240 to get -144.

$$\frac{76}{5}x-\frac{2}{5}x^{2}-144=0$$

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.

$$-\frac{2}{5}x^{2}+\frac{76}{5}x-144=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{2}{5} for a, \frac{76}{5} for b, and -144 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{76}{5}±\sqrt{\left(\frac{76}{5}\right)^{2}-4\left(-\frac{2}{5}\right)\left(-144\right)}}{2\left(-\frac{2}{5}\right)}$$

Square \frac{76}{5} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{76}{5}±\sqrt{\frac{5776}{25}-4\left(-\frac{2}{5}\right)\left(-144\right)}}{2\left(-\frac{2}{5}\right)}$$

Multiply -4 times -\frac{2}{5}.

$$x=\frac{-\frac{76}{5}±\sqrt{\frac{5776}{25}+\frac{8}{5}\left(-144\right)}}{2\left(-\frac{2}{5}\right)}$$

Multiply \frac{8}{5} times -144.

$$x=\frac{-\frac{76}{5}±\sqrt{\frac{5776}{25}-\frac{1152}{5}}}{2\left(-\frac{2}{5}\right)}$$

Add \frac{5776}{25} to -\frac{1152}{5} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{76}{5}±\sqrt{\frac{16}{25}}}{2\left(-\frac{2}{5}\right)}$$

Take the square root of \frac{16}{25}.

$$x=\frac{-\frac{76}{5}±\frac{4}{5}}{2\left(-\frac{2}{5}\right)}$$

Multiply 2 times -\frac{2}{5}.

$$x=\frac{-\frac{76}{5}±\frac{4}{5}}{-\frac{4}{5}}$$

Now solve the equation x=\frac{-\frac{76}{5}±\frac{4}{5}}{-\frac{4}{5}} when ± is plus. Add -\frac{76}{5} to \frac{4}{5} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{\frac{72}{5}}{-\frac{4}{5}}$$

Divide -\frac{72}{5} by -\frac{4}{5} by multiplying -\frac{72}{5} by the reciprocal of -\frac{4}{5}.

$$x=18$$

Now solve the equation x=\frac{-\frac{76}{5}±\frac{4}{5}}{-\frac{4}{5}} when ± is minus. Subtract \frac{4}{5} from -\frac{76}{5} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{16}{-\frac{4}{5}}$$

Divide -16 by -\frac{4}{5} by multiplying -16 by the reciprocal of -\frac{4}{5}.

$$x=20$$

The equation is now solved.

$$x=18 x=20$$

Steps for Completing the Square

Use the distributive property to multiply -\frac{2}{5}x+20 by 12+x and combine like terms.

$$\frac{76}{5}x-\frac{2}{5}x^{2}+240=384$$

Subtract 240 from both sides.

$$\frac{76}{5}x-\frac{2}{5}x^{2}=384-240$$

Subtract 240 from 384 to get 144.

$$\frac{76}{5}x-\frac{2}{5}x^{2}=144$$

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.

$$-\frac{2}{5}x^{2}+\frac{76}{5}x=144$$

Divide both sides of the equation by -\frac{2}{5}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{2}{5}x^{2}+\frac{76}{5}x}{-\frac{2}{5}}=\frac{144}{-\frac{2}{5}}$$

Dividing by -\frac{2}{5} undoes the multiplication by -\frac{2}{5}.

$$x^{2}+\frac{\frac{76}{5}}{-\frac{2}{5}}x=\frac{144}{-\frac{2}{5}}$$

Divide \frac{76}{5} by -\frac{2}{5} by multiplying \frac{76}{5} by the reciprocal of -\frac{2}{5}.

$$x^{2}-38x=\frac{144}{-\frac{2}{5}}$$

Divide 144 by -\frac{2}{5} by multiplying 144 by the reciprocal of -\frac{2}{5}.

$$x^{2}-38x=-360$$

Divide -38, the coefficient of the x term, by 2 to get -19. Then add the square of -19 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-38x+\left(-19\right)^{2}=-360+\left(-19\right)^{2}$$

Square -19.

$$x^{2}-38x+361=-360+361$$

Add -360 to 361.

$$x^{2}-38x+361=1$$

Factor x^{2}-38x+361. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-19\right)^{2}=1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-19\right)^{2}}=\sqrt{1}$$

Simplify.

$$x-19=1 x-19=-1$$

Add 19 to both sides of the equation.

$$x=20 x=18$$