Solve for x
x=5\sqrt{101}+45\approx 95.249378106
x=45-5\sqrt{101}\approx -5.249378106
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10x\left(x+10\right)\times 0.4+x\left(x+10\right)\times 20=\left(10x+100\right)\times 120+10x\times 120
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.
\left(10x^{2}+100x\right)\times 0.4+x\left(x+10\right)\times 20=\left(10x+100\right)\times 120+10x\times 120
Use the distributive property to multiply 10x by x+10.
4x^{2}+40x+x\left(x+10\right)\times 20=\left(10x+100\right)\times 120+10x\times 120
Use the distributive property to multiply 10x^{2}+100x by 0.4.
4x^{2}+40x+\left(x^{2}+10x\right)\times 20=\left(10x+100\right)\times 120+10x\times 120
Use the distributive property to multiply x by x+10.
4x^{2}+40x+20x^{2}+200x=\left(10x+100\right)\times 120+10x\times 120
Use the distributive property to multiply x^{2}+10x by 20.
24x^{2}+40x+200x=\left(10x+100\right)\times 120+10x\times 120
Combine 4x^{2} and 20x^{2} to get 24x^{2}.
24x^{2}+240x=\left(10x+100\right)\times 120+10x\times 120
Combine 40x and 200x to get 240x.
24x^{2}+240x=1200x+12000+10x\times 120
Use the distributive property to multiply 10x+100 by 120.
24x^{2}+240x=1200x+12000+1200x
Multiply 10 and 120 to get 1200.
24x^{2}+240x=2400x+12000
Combine 1200x and 1200x to get 2400x.
24x^{2}+240x-2400x=12000
Subtract 2400x from both sides.
24x^{2}-2160x=12000
Combine 240x and -2400x to get -2160x.
24x^{2}-2160x-12000=0
Subtract 12000 from both sides.
x=\frac{-\left(-2160\right)±\sqrt{\left(-2160\right)^{2}-4\times 24\left(-12000\right)}}{2\times 24}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 24 for a, -2160 for b, and -12000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-2160\right)±\sqrt{4665600-4\times 24\left(-12000\right)}}{2\times 24}
Square -2160.
x=\frac{-\left(-2160\right)±\sqrt{4665600-96\left(-12000\right)}}{2\times 24}
Multiply -4 times 24.
x=\frac{-\left(-2160\right)±\sqrt{4665600+1152000}}{2\times 24}
Multiply -96 times -12000.
x=\frac{-\left(-2160\right)±\sqrt{5817600}}{2\times 24}
Add 4665600 to 1152000.
x=\frac{-\left(-2160\right)±240\sqrt{101}}{2\times 24}
Take the square root of 5817600.
x=\frac{2160±240\sqrt{101}}{2\times 24}
The opposite of -2160 is 2160.
x=\frac{2160±240\sqrt{101}}{48}
Multiply 2 times 24.
x=\frac{240\sqrt{101}+2160}{48}
Now solve the equation x=\frac{2160±240\sqrt{101}}{48} when ± is plus. Add 2160 to 240\sqrt{101}.
x=5\sqrt{101}+45
Divide 2160+240\sqrt{101} by 48.
x=\frac{2160-240\sqrt{101}}{48}
Now solve the equation x=\frac{2160±240\sqrt{101}}{48} when ± is minus. Subtract 240\sqrt{101} from 2160.
x=45-5\sqrt{101}
Divide 2160-240\sqrt{101} by 48.
x=5\sqrt{101}+45 x=45-5\sqrt{101}
The equation is now solved.
10x\left(x+10\right)\times 0.4+x\left(x+10\right)\times 20=\left(10x+100\right)\times 120+10x\times 120
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.
\left(10x^{2}+100x\right)\times 0.4+x\left(x+10\right)\times 20=\left(10x+100\right)\times 120+10x\times 120
Use the distributive property to multiply 10x by x+10.
4x^{2}+40x+x\left(x+10\right)\times 20=\left(10x+100\right)\times 120+10x\times 120
Use the distributive property to multiply 10x^{2}+100x by 0.4.
4x^{2}+40x+\left(x^{2}+10x\right)\times 20=\left(10x+100\right)\times 120+10x\times 120
Use the distributive property to multiply x by x+10.
4x^{2}+40x+20x^{2}+200x=\left(10x+100\right)\times 120+10x\times 120
Use the distributive property to multiply x^{2}+10x by 20.
24x^{2}+40x+200x=\left(10x+100\right)\times 120+10x\times 120
Combine 4x^{2} and 20x^{2} to get 24x^{2}.
24x^{2}+240x=\left(10x+100\right)\times 120+10x\times 120
Combine 40x and 200x to get 240x.
24x^{2}+240x=1200x+12000+10x\times 120
Use the distributive property to multiply 10x+100 by 120.
24x^{2}+240x=1200x+12000+1200x
Multiply 10 and 120 to get 1200.
24x^{2}+240x=2400x+12000
Combine 1200x and 1200x to get 2400x.
24x^{2}+240x-2400x=12000
Subtract 2400x from both sides.
24x^{2}-2160x=12000
Combine 240x and -2400x to get -2160x.
\frac{24x^{2}-2160x}{24}=\frac{12000}{24}
Divide both sides by 24.
x^{2}+\left(-\frac{2160}{24}\right)x=\frac{12000}{24}
Dividing by 24 undoes the multiplication by 24.
x^{2}-90x=\frac{12000}{24}
Divide -2160 by 24.
x^{2}-90x=500
Divide 12000 by 24.
x^{2}-90x+\left(-45\right)^{2}=500+\left(-45\right)^{2}
Divide -90, the coefficient of the x term, by 2 to get -45. Then add the square of -45 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}-90x+2025=500+2025
Square -45.
x^{2}-90x+2025=2525
Add 500 to 2025.
\left(x-45\right)^{2}=2525
Factor x^{2}-90x+2025. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(x-45\right)^{2}}=\sqrt{2525}
Take the square root of both sides of the equation.
x-45=5\sqrt{101} x-45=-5\sqrt{101}
Simplify.
x=5\sqrt{101}+45 x=45-5\sqrt{101}
Add 45 to both sides of the equation.
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