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1=d+5\times \frac{1}{1000}d^{2}
Calculate 10 to the power of -3 and get \frac{1}{1000}.
1=d+\frac{1}{200}d^{2}
Multiply 5 and \frac{1}{1000} to get \frac{1}{200}.
d+\frac{1}{200}d^{2}=1
Swap sides so that all variable terms are on the left hand side.
d+\frac{1}{200}d^{2}-1=0
Subtract 1 from both sides.
\frac{1}{200}d^{2}+d-1=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.
d=\frac{-1±\sqrt{1^{2}-4\times \frac{1}{200}\left(-1\right)}}{2\times \frac{1}{200}}
This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{200} for a, 1 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
d=\frac{-1±\sqrt{1-4\times \frac{1}{200}\left(-1\right)}}{2\times \frac{1}{200}}
Square 1.
d=\frac{-1±\sqrt{1-\frac{1}{50}\left(-1\right)}}{2\times \frac{1}{200}}
Multiply -4 times \frac{1}{200}.
d=\frac{-1±\sqrt{1+\frac{1}{50}}}{2\times \frac{1}{200}}
Multiply -\frac{1}{50} times -1.
d=\frac{-1±\sqrt{\frac{51}{50}}}{2\times \frac{1}{200}}
Add 1 to \frac{1}{50}.
d=\frac{-1±\frac{\sqrt{102}}{10}}{2\times \frac{1}{200}}
Take the square root of \frac{51}{50}.
d=\frac{-1±\frac{\sqrt{102}}{10}}{\frac{1}{100}}
Multiply 2 times \frac{1}{200}.
d=\frac{\frac{\sqrt{102}}{10}-1}{\frac{1}{100}}
Now solve the equation d=\frac{-1±\frac{\sqrt{102}}{10}}{\frac{1}{100}} when ± is plus. Add -1 to \frac{\sqrt{102}}{10}.
d=10\sqrt{102}-100
Divide -1+\frac{\sqrt{102}}{10} by \frac{1}{100} by multiplying -1+\frac{\sqrt{102}}{10} by the reciprocal of \frac{1}{100}.
d=\frac{-\frac{\sqrt{102}}{10}-1}{\frac{1}{100}}
Now solve the equation d=\frac{-1±\frac{\sqrt{102}}{10}}{\frac{1}{100}} when ± is minus. Subtract \frac{\sqrt{102}}{10} from -1.
d=-10\sqrt{102}-100
Divide -1-\frac{\sqrt{102}}{10} by \frac{1}{100} by multiplying -1-\frac{\sqrt{102}}{10} by the reciprocal of \frac{1}{100}.
d=10\sqrt{102}-100 d=-10\sqrt{102}-100
The equation is now solved.
1=d+5\times \frac{1}{1000}d^{2}
Calculate 10 to the power of -3 and get \frac{1}{1000}.
1=d+\frac{1}{200}d^{2}
Multiply 5 and \frac{1}{1000} to get \frac{1}{200}.
d+\frac{1}{200}d^{2}=1
Swap sides so that all variable terms are on the left hand side.
\frac{1}{200}d^{2}+d=1
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{\frac{1}{200}d^{2}+d}{\frac{1}{200}}=\frac{1}{\frac{1}{200}}
Multiply both sides by 200.
d^{2}+\frac{1}{\frac{1}{200}}d=\frac{1}{\frac{1}{200}}
Dividing by \frac{1}{200} undoes the multiplication by \frac{1}{200}.
d^{2}+200d=\frac{1}{\frac{1}{200}}
Divide 1 by \frac{1}{200} by multiplying 1 by the reciprocal of \frac{1}{200}.
d^{2}+200d=200
Divide 1 by \frac{1}{200} by multiplying 1 by the reciprocal of \frac{1}{200}.
d^{2}+200d+100^{2}=200+100^{2}
Divide 200, the coefficient of the x term, by 2 to get 100. Then add the square of 100 to both sides of the equation. This step makes the left hand side of the equation a perfect square.
d^{2}+200d+10000=200+10000
Square 100.
d^{2}+200d+10000=10200
Add 200 to 10000.
\left(d+100\right)^{2}=10200
Factor d^{2}+200d+10000. 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(d+100\right)^{2}}=\sqrt{10200}
Take the square root of both sides of the equation.
d+100=10\sqrt{102} d+100=-10\sqrt{102}
Simplify.
d=10\sqrt{102}-100 d=-10\sqrt{102}-100
Subtract 100 from both sides of the equation.