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

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

Solve for x

$$x=18$$

$$x=20$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 5.

$$5\left(20-\frac{2x}{5}\right)\left(12+x\right)=1920$$

Use the distributive property to multiply 5 by 20-\frac{2x}{5}.

$$\left(100+5\left(-\frac{2x}{5}\right)\right)\left(12+x\right)=1920$$

Express 5\left(-\frac{2x}{5}\right) as a single fraction.

$$\left(100+\frac{-5\times 2x}{5}\right)\left(12+x\right)=1920$$

Cancel out 5 and 5.

$$\left(100-2x\right)\left(12+x\right)=1920$$

Apply the distributive property by multiplying each term of 100-2x by each term of 12+x.

$$1200+100x-24x-2x^{2}=1920$$

Combine 100x and -24x to get 76x.

$$1200+76x-2x^{2}=1920$$

Subtract 1920 from both sides.

$$1200+76x-2x^{2}-1920=0$$

Subtract 1920 from 1200 to get -720.

$$-720+76x-2x^{2}=0$$

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.

$$-2x^{2}+76x-720=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 76 for b, and -720 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-76±\sqrt{76^{2}-4\left(-2\right)\left(-720\right)}}{2\left(-2\right)}$$

Square 76.

$$x=\frac{-76±\sqrt{5776-4\left(-2\right)\left(-720\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-76±\sqrt{5776+8\left(-720\right)}}{2\left(-2\right)}$$

Multiply 8 times -720.

$$x=\frac{-76±\sqrt{5776-5760}}{2\left(-2\right)}$$

Add 5776 to -5760.

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

Take the square root of 16.

$$x=\frac{-76±4}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-76±4}{-4}$$

Now solve the equation x=\frac{-76±4}{-4} when ± is plus. Add -76 to 4.

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

Divide -72 by -4.

$$x=18$$

Now solve the equation x=\frac{-76±4}{-4} when ± is minus. Subtract 4 from -76.

$$x=-\frac{80}{-4}$$

Divide -80 by -4.

$$x=20$$

The equation is now solved.

$$x=18 x=20$$

Steps for Completing the Square

Multiply both sides of the equation by 5.

$$5\left(20-\frac{2x}{5}\right)\left(12+x\right)=1920$$

Use the distributive property to multiply 5 by 20-\frac{2x}{5}.

$$\left(100+5\left(-\frac{2x}{5}\right)\right)\left(12+x\right)=1920$$

Express 5\left(-\frac{2x}{5}\right) as a single fraction.

$$\left(100+\frac{-5\times 2x}{5}\right)\left(12+x\right)=1920$$

Cancel out 5 and 5.

$$\left(100-2x\right)\left(12+x\right)=1920$$

Apply the distributive property by multiplying each term of 100-2x by each term of 12+x.

$$1200+100x-24x-2x^{2}=1920$$

Combine 100x and -24x to get 76x.

$$1200+76x-2x^{2}=1920$$

Subtract 1200 from both sides.

$$76x-2x^{2}=1920-1200$$

Subtract 1200 from 1920 to get 720.

$$76x-2x^{2}=720$$

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.

$$-2x^{2}+76x=720$$

Divide both sides by -2.

$$\frac{-2x^{2}+76x}{-2}=\frac{720}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{76}{-2}x=\frac{720}{-2}$$

Divide 76 by -2.

$$x^{2}-38x=\frac{720}{-2}$$

Divide 720 by -2.

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

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.

$$x=20 x=18$$