Editing Quadratic equation (section)

===Avoiding loss of significance===
Although the quadratic formula provides an exact solution, the result is not exact if [[real number]]s are approximated during the computation, as usual in [[numerical analysis]], where real numbers are approximated by [[floating point number]]s (called "reals" in many [[programming language]]s). In this context, the quadratic formula is not completely [[numerical stability|stable]].

This occurs when the roots have different [[order of magnitude]], or, equivalently, when {{math|''b''<sup>2</sup>}} and  {{math|''b''<sup>2</sup> − 4''ac''}} are close in magnitude. In this case, the subtraction of two nearly equal numbers will cause [[loss of significance]] or [[catastrophic cancellation]] in the smaller root. To avoid this, the root that is smaller in magnitude, {{math|''r''}}, can be computed as <math>(c/a)/R</math> where {{math|''R''}} is the root that is bigger in magnitude. This is equivalent to using the formula

<math display="block">x =\frac{-2c}{b \pm \sqrt {b^2-4ac}}</math>

using the plus sign if <math>b>0</math> and the minus sign if <math>b<0.</math>

A second form of cancellation can occur between the terms {{math|''b''<sup>2</sup>}} and {{math|4''ac''}} of the discriminant, that is when the two roots are very close. This can lead to loss of up to half of correct significant figures in the roots.<ref name="kahan"/><ref name="Higham2002">{{Citation |first=Nicholas |last=Higham |title=Accuracy and Stability of Numerical Algorithms |edition=2nd |publisher=SIAM |year=2002 |isbn=978-0-89871-521-7 |page=10 }}</ref>