Solve 180(30+7x)(1.8-x/10)+120(1.5-x/20)(50-7x)=18720

$$180(30+7x)(1.8- \frac{ x }{ 10 } )+120(1.5- \frac{ x }{ 20 } )(50-7x)=18720$$

Solve for x

$$x=2$$

$$x=0$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 20, the least common multiple of 10,20.

$$3600\left(30+7x\right)\left(1.8-\frac{x}{10}\right)+2400\left(1.5-\frac{x}{20}\right)\left(50-7x\right)=374400$$

Use the distributive property to multiply 3600 by 30+7x.

$$\left(108000+25200x\right)\left(1.8-\frac{x}{10}\right)+2400\left(1.5-\frac{x}{20}\right)\left(50-7x\right)=374400$$

Apply the distributive property by multiplying each term of 108000+25200x by each term of 1.8-\frac{x}{10}.

$$194400+108000\left(-\frac{x}{10}\right)+45360x+25200x\left(-\frac{x}{10}\right)+2400\left(1.5-\frac{x}{20}\right)\left(50-7x\right)=374400$$

Cancel out 10, the greatest common factor in 108000 and 10.

$$194400-10800x+45360x+25200x\left(-\frac{x}{10}\right)+2400\left(1.5-\frac{x}{20}\right)\left(50-7x\right)=374400$$

Combine -10800x and 45360x to get 34560x.

$$194400+34560x+25200x\left(-\frac{x}{10}\right)+2400\left(1.5-\frac{x}{20}\right)\left(50-7x\right)=374400$$

Cancel out 10, the greatest common factor in 25200 and 10.

$$194400+34560x-2520xx+2400\left(1.5-\frac{x}{20}\right)\left(50-7x\right)=374400$$

Multiply x and x to get x^{2}.

$$194400+34560x-2520x^{2}+2400\left(1.5-\frac{x}{20}\right)\left(50-7x\right)=374400$$

Use the distributive property to multiply 2400 by 1.5-\frac{x}{20}.

$$194400+34560x-2520x^{2}+\left(3600+2400\left(-\frac{x}{20}\right)\right)\left(50-7x\right)=374400$$

Cancel out 20, the greatest common factor in 2400 and 20.

$$194400+34560x-2520x^{2}+\left(3600-120x\right)\left(50-7x\right)=374400$$

Apply the distributive property by multiplying each term of 3600-120x by each term of 50-7x.

$$194400+34560x-2520x^{2}[phone]x-6000x+840x^{2}=374400$$

Combine -25200x and -6000x to get -31200x.

$$194400+34560x-2520x^{2}[phone]x+840x^{2}=374400$$

Add 194400 and 180000 to get 374400.

$$374400+34560x-2520x^{2}-31200x+840x^{2}=374400$$

Combine 34560x and -31200x to get 3360x.

$$374400+3360x-2520x^{2}+840x^{2}=374400$$

Combine -2520x^{2} and 840x^{2} to get -1680x^{2}.

$$374400+3360x-1680x^{2}=374400$$

Subtract 374400 from both sides.

$$374400+3360x-1680x^{2}-374400=0$$

Subtract 374400 from 374400 to get 0.

$$3360x-1680x^{2}=0$$

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.

$$-1680x^{2}+3360x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1680 for a, 3360 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3360±\sqrt{3360^{2}}}{2\left(-1680\right)}$$

Take the square root of 3360^{2}.

$$x=\frac{-3360±3360}{2\left(-1680\right)}$$

Multiply 2 times -1680.

$$x=\frac{-3360±3360}{-3360}$$

Now solve the equation x=\frac{-3360±3360}{-3360} when ± is plus. Add -3360 to 3360.

$$x=\frac{0}{-3360}$$

Divide 0 by -3360.

$$x=0$$

Now solve the equation x=\frac{-3360±3360}{-3360} when ± is minus. Subtract 3360 from -3360.

$$x=-\frac{6720}{-3360}$$

Divide -6720 by -3360.

$$x=2$$

The equation is now solved.

$$x=0 x=2$$

Steps for Completing the Square

Multiply both sides of the equation by 20, the least common multiple of 10,20.

$$3600\left(30+7x\right)\left(1.8-\frac{x}{10}\right)+2400\left(1.5-\frac{x}{20}\right)\left(50-7x\right)=374400$$

Use the distributive property to multiply 3600 by 30+7x.

$$\left(108000+25200x\right)\left(1.8-\frac{x}{10}\right)+2400\left(1.5-\frac{x}{20}\right)\left(50-7x\right)=374400$$

Apply the distributive property by multiplying each term of 108000+25200x by each term of 1.8-\frac{x}{10}.

$$194400+108000\left(-\frac{x}{10}\right)+45360x+25200x\left(-\frac{x}{10}\right)+2400\left(1.5-\frac{x}{20}\right)\left(50-7x\right)=374400$$

Cancel out 10, the greatest common factor in 108000 and 10.

$$194400-10800x+45360x+25200x\left(-\frac{x}{10}\right)+2400\left(1.5-\frac{x}{20}\right)\left(50-7x\right)=374400$$

Combine -10800x and 45360x to get 34560x.

$$194400+34560x+25200x\left(-\frac{x}{10}\right)+2400\left(1.5-\frac{x}{20}\right)\left(50-7x\right)=374400$$

Cancel out 10, the greatest common factor in 25200 and 10.

$$194400+34560x-2520xx+2400\left(1.5-\frac{x}{20}\right)\left(50-7x\right)=374400$$

Multiply x and x to get x^{2}.

$$194400+34560x-2520x^{2}+2400\left(1.5-\frac{x}{20}\right)\left(50-7x\right)=374400$$

Use the distributive property to multiply 2400 by 1.5-\frac{x}{20}.

$$194400+34560x-2520x^{2}+\left(3600+2400\left(-\frac{x}{20}\right)\right)\left(50-7x\right)=374400$$

Cancel out 20, the greatest common factor in 2400 and 20.

$$194400+34560x-2520x^{2}+\left(3600-120x\right)\left(50-7x\right)=374400$$

Apply the distributive property by multiplying each term of 3600-120x by each term of 50-7x.

$$194400+34560x-2520x^{2}[phone]x-6000x+840x^{2}=374400$$

Combine -25200x and -6000x to get -31200x.

$$194400+34560x-2520x^{2}[phone]x+840x^{2}=374400$$

Add 194400 and 180000 to get 374400.

$$374400+34560x-2520x^{2}-31200x+840x^{2}=374400$$

Combine 34560x and -31200x to get 3360x.

$$374400+3360x-2520x^{2}+840x^{2}=374400$$

Combine -2520x^{2} and 840x^{2} to get -1680x^{2}.

$$374400+3360x-1680x^{2}=374400$$

Subtract 374400 from both sides.

$$3360x-1680x^{2}=374400-374400$$

Subtract 374400 from 374400 to get 0.

$$3360x-1680x^{2}=0$$

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.

$$-1680x^{2}+3360x=0$$

Divide both sides by -1680.

$$\frac{-1680x^{2}+3360x}{-1680}=\frac{0}{-1680}$$

Dividing by -1680 undoes the multiplication by -1680.

$$x^{2}+\frac{3360}{-1680}x=\frac{0}{-1680}$$

Divide 3360 by -1680.

$$x^{2}-2x=\frac{0}{-1680}$$

Divide 0 by -1680.

$$x^{2}-2x=0$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

Add 1 to both sides of the equation.

$$x=2 x=0$$