Experimental Error on a quadratic

[dustydude]

Hello,
I'm looking to derive the error of a quadratic equation as a function of x.

y=ax²+bx+c

y is measured and variables a,b,c have an associated error Da,Db,Dc

To solve for x you complete the square.

$$x=&\pm\sqrt{\dfrac{y-c}{a}+\dfrac{b^{2}}{4a^{2}}}\mp\dfrac{b}{2a}$$

If the variables y,c and b had an assoicated error Dy,Dc and Db.

Then the error would be Dx squared would be the sum of partial derivatives with respect to each variable times by the error on the variable all squared?
$$\left(\Delta x\,\right)^{2}=\left(\dfrac{\partial x}{\partial a}\Delta a\right)^{2}+\left(\dfrac{\partial x}{\partial b}\Delta b\right)^{2}+...$$

It seems quite hairy to go though and I was curious if anyone knew a better way to do it or anyone else who has done it?

 

[Bob S]

The formula is a little more complex if there are any correlations between the individual uncertainties Δa, Δb, and Δc. See

http://mathworld.wolfram.com/CorrelationCoefficient.html

http://mathworld.wolfram.com/LeastSquaresFitting.html

Bob S

 

[dustydude]

Thanks for the reply.

If i understand what your getting at then the method in the links are used when fitting a curve to data points.

The only measured value is y and I am looking for the error in x.
a,b,c have one variable in common but also other variables. All the variables have an Error associated with them.

 

[Dale]

If x is a function of a, b, c, and y, and if all four (a, b, c, y) have some error associated with them then the error of x will be given by the "hairy" propagation of errors formula that you posted. If you have access to Mathematica or Maple it should not be very difficult to do, but don't expect it to simplify too much, particularly if there are any correlations in the errors.

 

[dustydude]

Thanks for the comments ppl!