Solve for x
x=\frac{360\sqrt{170}}{31}-40.2\approx 111.213733282
x=-\frac{360\sqrt{170}}{31}-40.2\approx -191.613733282
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1780.4\times 10000=128\times 10^{4}+28.83\times 10^{2}\left(\frac{x}{2}+20.1\right)^{2}
Calculate 10 to the power of 4 and get 10000.
17804000=128\times 10^{4}+28.83\times 10^{2}\left(\frac{x}{2}+20.1\right)^{2}
Multiply 1780.4 and 10000 to get 17804000.
17804000=128\times 10000+28.83\times 10^{2}\left(\frac{x}{2}+20.1\right)^{2}
Calculate 10 to the power of 4 and get 10000.
17804000=1280000+28.83\times 10^{2}\left(\frac{x}{2}+20.1\right)^{2}
Multiply 128 and 10000 to get 1280000.
17804000=1280000+28.83\times 100\left(\frac{x}{2}+20.1\right)^{2}
Calculate 10 to the power of 2 and get 100.
17804000=1280000+2883\left(\frac{x}{2}+20.1\right)^{2}
Multiply 28.83 and 100 to get 2883.
17804000=1280000+2883\left(\left(\frac{x}{2}\right)^{2}+40.2\times \frac{x}{2}+404.01\right)
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\frac{x}{2}+20.1\right)^{2}.
17804000=1280000+2883\left(\frac{x^{2}}{2^{2}}+40.2\times \frac{x}{2}+404.01\right)
To raise \frac{x}{2} to a power, raise both numerator and denominator to the power and then divide.
17804000=1280000+2883\times \frac{x^{2}}{2^{2}}+115896.6\times \frac{x}{2}+1164760.83
Use the distributive property to multiply 2883 by \frac{x^{2}}{2^{2}}+40.2\times \frac{x}{2}+404.01.
17804000=1280000+2883\times \frac{x^{2}}{4}+115896.6\times \frac{x}{2}+1164760.83
Calculate 2 to the power of 2 and get 4.
17804000=1280000+\frac{2883x^{2}}{4}+115896.6\times \frac{x}{2}+1164760.83
Express 2883\times \frac{x^{2}}{4} as a single fraction.
17804000=2444760.83+\frac{2883x^{2}}{4}+115896.6\times \frac{x}{2}
Add 1280000 and 1164760.83 to get 2444760.83.
2444760.83+\frac{2883x^{2}}{4}+115896.6\times \frac{x}{2}=17804000
Swap sides so that all variable terms are on the left hand side.
2444760.83+\frac{2883x^{2}}{4}+115896.6\times \frac{x}{2}-17804000=0
Subtract 17804000 from both sides.
-15359239.17+\frac{2883x^{2}}{4}+115896.6\times \frac{x}{2}=0
Subtract 17804000 from 2444760.83 to get -15359239.17.
-61436956.68+2883x^{2}+231793.2x=0
Multiply both sides of the equation by 4, the least common multiple of 4,2.
2883x^{2}+231793.2x-61436956.68=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.
x=\frac{-231793.2±\sqrt{231793.2^{2}-4\times 2883\left(-61436956.68\right)}}{2\times 2883}
This equation is in standard form: ax^{2}+bx+c=0. Substitute 2883 for a, 231793.2 for b, and -61436956.68 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-231793.2±\sqrt{53728087566.24-4\times 2883\left(-61436956.68\right)}}{2\times 2883}
Square 231793.2 by squaring both the numerator and the denominator of the fraction.
x=\frac{-231793.2±\sqrt{53728087566.24-11532\left(-61436956.68\right)}}{2\times 2883}
Multiply -4 times 2883.
x=\frac{-231793.2±\sqrt{\frac{1343202189156+17712274610844}{25}}}{2\times 2883}
Multiply -11532 times -61436956.68.
x=\frac{-231793.2±\sqrt{762219072000}}{2\times 2883}
Add 53728087566.24 to 708490984433.76 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
x=\frac{-231793.2±66960\sqrt{170}}{2\times 2883}
Take the square root of 762219072000.
x=\frac{-231793.2±66960\sqrt{170}}{5766}
Multiply 2 times 2883.
x=\frac{66960\sqrt{170}-231793.2}{5766}
Now solve the equation x=\frac{-231793.2±66960\sqrt{170}}{5766} when ± is plus. Add -231793.2 to 66960\sqrt{170}.
x=\frac{360\sqrt{170}}{31}-\frac{201}{5}
Divide -231793.2+66960\sqrt{170} by 5766.
x=\frac{-66960\sqrt{170}-231793.2}{5766}
Now solve the equation x=\frac{-231793.2±66960\sqrt{170}}{5766} when ± is minus. Subtract 66960\sqrt{170} from -231793.2.
x=-\frac{360\sqrt{170}}{31}-\frac{201}{5}
Divide -231793.2-66960\sqrt{170} by 5766.
x=\frac{360\sqrt{170}}{31}-\frac{201}{5} x=-\frac{360\sqrt{170}}{31}-\frac{201}{5}
The equation is now solved.
1780.4\times 10000=128\times 10^{4}+28.83\times 10^{2}\left(\frac{x}{2}+20.1\right)^{2}
Calculate 10 to the power of 4 and get 10000.
17804000=128\times 10^{4}+28.83\times 10^{2}\left(\frac{x}{2}+20.1\right)^{2}
Multiply 1780.4 and 10000 to get 17804000.
17804000=128\times 10000+28.83\times 10^{2}\left(\frac{x}{2}+20.1\right)^{2}
Calculate 10 to the power of 4 and get 10000.
17804000=1280000+28.83\times 10^{2}\left(\frac{x}{2}+20.1\right)^{2}
Multiply 128 and 10000 to get 1280000.
17804000=1280000+28.83\times 100\left(\frac{x}{2}+20.1\right)^{2}
Calculate 10 to the power of 2 and get 100.
17804000=1280000+2883\left(\frac{x}{2}+20.1\right)^{2}
Multiply 28.83 and 100 to get 2883.
17804000=1280000+2883\left(\left(\frac{x}{2}\right)^{2}+40.2\times \frac{x}{2}+404.01\right)
Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\frac{x}{2}+20.1\right)^{2}.
17804000=1280000+2883\left(\frac{x^{2}}{2^{2}}+40.2\times \frac{x}{2}+404.01\right)
To raise \frac{x}{2} to a power, raise both numerator and denominator to the power and then divide.
17804000=1280000+2883\times \frac{x^{2}}{2^{2}}+115896.6\times \frac{x}{2}+1164760.83
Use the distributive property to multiply 2883 by \frac{x^{2}}{2^{2}}+40.2\times \frac{x}{2}+404.01.
17804000=1280000+2883\times \frac{x^{2}}{4}+115896.6\times \frac{x}{2}+1164760.83
Calculate 2 to the power of 2 and get 4.
17804000=1280000+\frac{2883x^{2}}{4}+115896.6\times \frac{x}{2}+1164760.83
Express 2883\times \frac{x^{2}}{4} as a single fraction.
17804000=2444760.83+\frac{2883x^{2}}{4}+115896.6\times \frac{x}{2}
Add 1280000 and 1164760.83 to get 2444760.83.
2444760.83+\frac{2883x^{2}}{4}+115896.6\times \frac{x}{2}=17804000
Swap sides so that all variable terms are on the left hand side.
\frac{2883x^{2}}{4}+115896.6\times \frac{x}{2}=17804000-2444760.83
Subtract 2444760.83 from both sides.
\frac{2883x^{2}}{4}+115896.6\times \frac{x}{2}=15359239.17
Subtract 2444760.83 from 17804000 to get 15359239.17.
2883x^{2}+231793.2x=61436956.68
Multiply both sides of the equation by 4, the least common multiple of 4,2.
\frac{2883x^{2}+231793.2x}{2883}=\frac{61436956.68}{2883}
Divide both sides by 2883.
x^{2}+\frac{231793.2}{2883}x=\frac{61436956.68}{2883}
Dividing by 2883 undoes the multiplication by 2883.
x^{2}+80.4x=\frac{61436956.68}{2883}
Divide 231793.2 by 2883.
x^{2}+80.4x=\frac{511974639}{24025}
Divide 61436956.68 by 2883.
x^{2}+80.4x+40.2^{2}=\frac{511974639}{24025}+40.2^{2}
Divide 80.4, the coefficient of the x term, by 2 to get 40.2. Then add the square of 40.2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+80.4x+1616.04=\frac{511974639}{24025}+1616.04
Square 40.2 by squaring both the numerator and the denominator of the fraction.
x^{2}+80.4x+1616.04=\frac{22032000}{961}
Add \frac{511974639}{24025} to 1616.04 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.
\left(x+40.2\right)^{2}=\frac{22032000}{961}
Factor x^{2}+80.4x+1616.04. 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+40.2\right)^{2}}=\sqrt{\frac{22032000}{961}}
Take the square root of both sides of the equation.
x+40.2=\frac{360\sqrt{170}}{31} x+40.2=-\frac{360\sqrt{170}}{31}
Simplify.
x=\frac{360\sqrt{170}}{31}-\frac{201}{5} x=-\frac{360\sqrt{170}}{31}-\frac{201}{5}
Subtract 40.2 from both sides of the equation.
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