Quadratic Formula Rewritten -- by Harry J. Smith

The standard quadratic formula is:

It gives the roots of the quadratic equation:

But if the magnitude of

Is small compared to

then one of the real roots will not be computed accurately when using floating point arithmetic in a computer program.

The reason for this is that when you subtract two numbers that are almost equal using floating point arithmetic, you have very few significant bits left in the resulting mantissa.

To fix this problem, first ensure that

If it is not, change the sign of all three coefficients:

The first real root can now be computed by

To get the second root

Multiply the numerator and denominator of this by

This gives

This is an accurate way to compute the second real root. It even works when

-Harry

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