Solve 04+20/10=120/x+120/x+10

$$04 + \frac { 20 } { 10 } = \frac { 120 } { x } + \frac { 120 } { x + 10 }$$

Solve for x

$$x=5\sqrt{145}+55\approx 115.207972894$$

$$x=55-5\sqrt{145}\approx -5.207972894$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -10,0 since division by zero is not defined. Multiply both sides of the equation by 10x\left(x+10\right), the least common multiple of 10,x,x+10.

$$0\times 4\times 10x\left(x+10\right)+x\left(x+10\right)\times 20=\left(10x+100\right)\times 120+10x\times 120$$

Multiply 0 and 4 to get 0.

$$0\times 10x\left(x+10\right)+x\left(x+10\right)\times 20=\left(10x+100\right)\times 120+10x\times 120$$

Multiply 0 and 10 to get 0.

$$0x\left(x+10\right)+x\left(x+10\right)\times 20=\left(10x+100\right)\times 120+10x\times 120$$

Anything times zero gives zero.

$$0+x\left(x+10\right)\times 20=\left(10x+100\right)\times 120+10x\times 120$$

Use the distributive property to multiply x by x+10.

$$0+\left(x^{2}+10x\right)\times 20=\left(10x+100\right)\times 120+10x\times 120$$

Use the distributive property to multiply x^{2}+10x by 20.

$$0+20x^{2}+200x=\left(10x+100\right)\times 120+10x\times 120$$

Anything plus zero gives itself.

$$20x^{2}+200x=\left(10x+100\right)\times 120+10x\times 120$$

Use the distributive property to multiply 10x+100 by 120.

$$20x^{2}+200x=1200x+12000+10x\times 120$$

Multiply 10 and 120 to get 1200.

$$20x^{2}+200x=1200x+12000+1200x$$

Combine 1200x and 1200x to get 2400x.

$$20x^{2}+200x=2400x+12000$$

Subtract 2400x from both sides.

$$20x^{2}+200x-2400x=12000$$

Combine 200x and -2400x to get -2200x.

$$20x^{2}-2200x=12000$$

Subtract 12000 from both sides.

$$20x^{2}-2200x-12000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 20 for a, -2200 for b, and -12000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2200\right)±\sqrt{\left(-2200\right)^{2}-4\times 20\left(-12000\right)}}{2\times 20}$$

Square -2200.

$$x=\frac{-\left(-2200\right)±\sqrt{4840000-4\times 20\left(-12000\right)}}{2\times 20}$$

Multiply -4 times 20.

$$x=\frac{-\left(-2200\right)±\sqrt{4840000-80\left(-12000\right)}}{2\times 20}$$

Multiply -80 times -12000.

$$x=\frac{-\left(-2200\right)±\sqrt{4840000+960000}}{2\times 20}$$

Add 4840000 to 960000.

$$x=\frac{-\left(-2200\right)±\sqrt{5800000}}{2\times 20}$$

Take the square root of 5800000.

$$x=\frac{-\left(-2200\right)±200\sqrt{145}}{2\times 20}$$

The opposite of -2200 is 2200.

$$x=\frac{2200±200\sqrt{145}}{2\times 20}$$

Multiply 2 times 20.

$$x=\frac{2200±200\sqrt{145}}{40}$$

Now solve the equation x=\frac{2200±200\sqrt{145}}{40} when ± is plus. Add 2200 to 200\sqrt{145}.

$$x=\frac{200\sqrt{145}+2200}{40}$$

Divide 2200+200\sqrt{145} by 40.

$$x=5\sqrt{145}+55$$

Now solve the equation x=\frac{2200±200\sqrt{145}}{40} when ± is minus. Subtract 200\sqrt{145} from 2200.

$$x=\frac{2200-200\sqrt{145}}{40}$$

Divide 2200-200\sqrt{145} by 40.

$$x=55-5\sqrt{145}$$

The equation is now solved.

$$x=5\sqrt{145}+55 x=55-5\sqrt{145}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -10,0 since division by zero is not defined. Multiply both sides of the equation by 10x\left(x+10\right), the least common multiple of 10,x,x+10.

$$0\times 4\times 10x\left(x+10\right)+x\left(x+10\right)\times 20=\left(10x+100\right)\times 120+10x\times 120$$

Multiply 0 and 4 to get 0.

$$0\times 10x\left(x+10\right)+x\left(x+10\right)\times 20=\left(10x+100\right)\times 120+10x\times 120$$

Multiply 0 and 10 to get 0.

$$0x\left(x+10\right)+x\left(x+10\right)\times 20=\left(10x+100\right)\times 120+10x\times 120$$

Anything times zero gives zero.

$$0+x\left(x+10\right)\times 20=\left(10x+100\right)\times 120+10x\times 120$$

Use the distributive property to multiply x by x+10.

$$0+\left(x^{2}+10x\right)\times 20=\left(10x+100\right)\times 120+10x\times 120$$

Use the distributive property to multiply x^{2}+10x by 20.

$$0+20x^{2}+200x=\left(10x+100\right)\times 120+10x\times 120$$

Anything plus zero gives itself.

$$20x^{2}+200x=\left(10x+100\right)\times 120+10x\times 120$$

Use the distributive property to multiply 10x+100 by 120.

$$20x^{2}+200x=1200x+12000+10x\times 120$$

Multiply 10 and 120 to get 1200.

$$20x^{2}+200x=1200x+12000+1200x$$

Combine 1200x and 1200x to get 2400x.

$$20x^{2}+200x=2400x+12000$$

Subtract 2400x from both sides.

$$20x^{2}+200x-2400x=12000$$

Combine 200x and -2400x to get -2200x.

$$20x^{2}-2200x=12000$$

Divide both sides by 20.

$$\frac{20x^{2}-2200x}{20}=\frac{12000}{20}$$

Dividing by 20 undoes the multiplication by 20.

$$x^{2}+\left(-\frac{2200}{20}\right)x=\frac{12000}{20}$$

Divide -2200 by 20.

$$x^{2}-110x=\frac{12000}{20}$$

Divide 12000 by 20.

$$x^{2}-110x=600$$

Divide -110, the coefficient of the x term, by 2 to get -55. Then add the square of -55 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-110x+\left(-55\right)^{2}=600+\left(-55\right)^{2}$$

Square -55.

$$x^{2}-110x+3025=600+3025$$

Add 600 to 3025.

$$x^{2}-110x+3025=3625$$

Factor x^{2}-110x+3025. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-55\right)^{2}=3625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-55\right)^{2}}=\sqrt{3625}$$

Simplify.

$$x-55=5\sqrt{145} x-55=-5\sqrt{145}$$

Add 55 to both sides of the equation.

$$x=5\sqrt{145}+55 x=55-5\sqrt{145}$$