Solve for x
x=\frac{-5\sqrt{\left(Re(C)-9048.2\right)\left(Re(C)+9047.8\right)}+5Re(C)-1}{290}
x=\frac{5\sqrt{\left(Re(C)-9048.2\right)\left(Re(C)+9047.8\right)}+5Re(C)-1}{290}
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Re(C)x-705744=0.2x+29x^{2}
Subtract 705744 from both sides.
Re(C)x-705744-0.2x=29x^{2}
Subtract 0.2x from both sides.
Re(C)x-705744-0.2x-29x^{2}=0
Subtract 29x^{2} from both sides.
\left(Re(C)-0.2\right)x-705744-29x^{2}=0
Combine all terms containing x.
-29x^{2}+\left(Re(C)-0.2\right)x-705744=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{-\left(Re(C)-0.2\right)±\sqrt{\left(Re(C)-0.2\right)^{2}-4\left(-29\right)\left(-705744\right)}}{2\left(-29\right)}
This equation is in standard form: ax^{2}+bx+c=0. Substitute -29 for a, Re(C)-0.2 for b, and -705744 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(Re(C)-0.2\right)±\sqrt{\left(Re(C)-0.2\right)^{2}+116\left(-705744\right)}}{2\left(-29\right)}
Multiply -4 times -29.
x=\frac{-\left(Re(C)-0.2\right)±\sqrt{\left(Re(C)-0.2\right)^{2}-81866304}}{2\left(-29\right)}
Multiply 116 times -705744.
x=\frac{-\left(Re(C)-0.2\right)±\sqrt{\left(Re(C)-9048.2\right)\left(Re(C)+9047.8\right)}}{2\left(-29\right)}
Add \left(Re(C)-0.2\right)^{2} to -81866304.
x=\frac{-Re(C)+\frac{1}{5}±\sqrt{\left(Re(C)-9048.2\right)\left(Re(C)+9047.8\right)}}{-58}
Multiply 2 times -29.
x=\frac{\sqrt{\left(Re(C)-9048.2\right)\left(Re(C)+9047.8\right)}-Re(C)+\frac{1}{5}}{-58}
Now solve the equation x=\frac{-Re(C)+\frac{1}{5}±\sqrt{\left(Re(C)-9048.2\right)\left(Re(C)+9047.8\right)}}{-58} when ± is plus. Add -Re(C)+\frac{1}{5} to \sqrt{\left(Re(C)+9047.8\right)\left(Re(C)-9048.2\right)}.
x=-\frac{\sqrt{\left(Re(C)-9048.2\right)\left(Re(C)+9047.8\right)}}{58}+\frac{Re(C)}{58}-\frac{1}{290}
Divide -Re(C)+\frac{1}{5}+\sqrt{\left(Re(C)+9047.8\right)\left(Re(C)-9048.2\right)} by -58.
x=\frac{-\sqrt{\left(Re(C)-9048.2\right)\left(Re(C)+9047.8\right)}-Re(C)+\frac{1}{5}}{-58}
Now solve the equation x=\frac{-Re(C)+\frac{1}{5}±\sqrt{\left(Re(C)-9048.2\right)\left(Re(C)+9047.8\right)}}{-58} when ± is minus. Subtract \sqrt{\left(Re(C)+9047.8\right)\left(Re(C)-9048.2\right)} from -Re(C)+\frac{1}{5}.
x=\frac{\sqrt{\left(Re(C)-9048.2\right)\left(Re(C)+9047.8\right)}}{58}+\frac{Re(C)}{58}-\frac{1}{290}
Divide -Re(C)+\frac{1}{5}-\sqrt{\left(Re(C)+9047.8\right)\left(Re(C)-9048.2\right)} by -58.
x=-\frac{\sqrt{\left(Re(C)-9048.2\right)\left(Re(C)+9047.8\right)}}{58}+\frac{Re(C)}{58}-\frac{1}{290} x=\frac{\sqrt{\left(Re(C)-9048.2\right)\left(Re(C)+9047.8\right)}}{58}+\frac{Re(C)}{58}-\frac{1}{290}
The equation is now solved.
Re(C)x-0.2x=705744+29x^{2}
Subtract 0.2x from both sides.
Re(C)x-0.2x-29x^{2}=705744
Subtract 29x^{2} from both sides.
\left(Re(C)-0.2\right)x-29x^{2}=705744
Combine all terms containing x.
-29x^{2}+\left(Re(C)-0.2\right)x=705744
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.
\frac{-29x^{2}+\left(Re(C)-0.2\right)x}{-29}=\frac{705744}{-29}
Divide both sides by -29.
x^{2}+\frac{Re(C)-0.2}{-29}x=\frac{705744}{-29}
Dividing by -29 undoes the multiplication by -29.
x^{2}+\left(-\frac{Re(C)}{29}+\frac{1}{145}\right)x=\frac{705744}{-29}
Divide Re(C)-0.2 by -29.
x^{2}+\left(-\frac{Re(C)}{29}+\frac{1}{145}\right)x=-24336
Divide 705744 by -29.
x^{2}+\left(-\frac{Re(C)}{29}+\frac{1}{145}\right)x+\left(-\frac{Re(C)}{58}+\frac{1}{290}\right)^{2}=-24336+\left(-\frac{Re(C)}{58}+\frac{1}{290}\right)^{2}
Divide -\frac{Re(C)}{29}+\frac{1}{145}, the coefficient of the x term, by 2 to get -\frac{Re(C)}{58}+\frac{1}{290}. Then add the square of -\frac{Re(C)}{58}+\frac{1}{290} to both sides of the equation. This step makes the left hand side of the equation a perfect square.
x^{2}+\left(-\frac{Re(C)}{29}+\frac{1}{145}\right)x+\frac{\left(-5Re(C)+1\right)^{2}}{84100}=-24336+\frac{\left(-5Re(C)+1\right)^{2}}{84100}
Square -\frac{Re(C)}{58}+\frac{1}{290}.
x^{2}+\left(-\frac{Re(C)}{29}+\frac{1}{145}\right)x+\frac{\left(-5Re(C)+1\right)^{2}}{84100}=\frac{\left(Re(C)\right)^{2}}{3364}-\frac{Re(C)}{8410}-\frac{2046657599}{84100}
Add -24336 to \frac{\left(-5Re(C)+1\right)^{2}}{84100}.
\left(x-\frac{Re(C)}{58}+\frac{1}{290}\right)^{2}=\frac{\left(Re(C)\right)^{2}}{3364}-\frac{Re(C)}{8410}-\frac{2046657599}{84100}
Factor x^{2}+\left(-\frac{Re(C)}{29}+\frac{1}{145}\right)x+\frac{\left(-5Re(C)+1\right)^{2}}{84100}. 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-\frac{Re(C)}{58}+\frac{1}{290}\right)^{2}}=\sqrt{\frac{\left(Re(C)\right)^{2}}{3364}-\frac{Re(C)}{8410}-\frac{2046657599}{84100}}
Take the square root of both sides of the equation.
x-\frac{Re(C)}{58}+\frac{1}{290}=\frac{\sqrt{25\left(Re(C)\right)^{2}-10Re(C)-2046657599}}{290} x-\frac{Re(C)}{58}+\frac{1}{290}=-\frac{\sqrt{25\left(Re(C)\right)^{2}-10Re(C)-2046657599}}{290}
Simplify.
x=\frac{\sqrt{25\left(Re(C)\right)^{2}-10Re(C)-2046657599}}{290}+\frac{Re(C)}{58}-\frac{1}{290} x=-\frac{\sqrt{25\left(Re(C)\right)^{2}-10Re(C)-2046657599}}{290}+\frac{Re(C)}{58}-\frac{1}{290}
Subtract -\frac{Re(C)}{58}+\frac{1}{290} from both sides of the equation.
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