Circuit Theory/Quadratic Equation Revisited

The Quadratic EquationEdit

The roots to the quadratic polynomial

 

are easily derived and many people memorized them in high school:

 

Derivation of the Quadratic EquationEdit

To derive this set   and complete the square:

 

Solving for   gives

 

Taking the square root of both sides and putting everything over a common denominator gives

 

Numerical Instability of the Usual Formulation of the Quadratic EquationEdit

Middlebrook has pointed out that this is a poor expression from a numerical point of view for certain values of  ,  , and  .

[Give an example here]

Middlebrook showed how a better expression can be obtained as follows. First, factor   out of the expression:

 


Now let

 

Then

 

A More Numerically Stable Formulation of the Negative RootEdit

Considering just the negative square root we have

 

Multiplying the numerator and denominator by   gives

 

By defining

 

we can write

 

Note that as  ,  .

Finding the Positive Root Using the Same ApproachEdit

Turning now to the positive square root we have

 

Using the two roots   and  , we can factor the quadratic equation

 

Accuracy for Low Edit

For values of   the value of   is within 10% of 1 and we may neglect it. As noted above, the approximation gets better as  . With this approximation the quadratic equation has a very simple factorization:

 

an expression that involves no messy square roots and can be written by inspection. Of course, it is necessary to check the assumption about   being small before using the simplification. Without this simplification,   needs to be calculated and the roots are slightly more complicated.

[Explore the consequences if  .]