Editing Quadratic equation (section)

===Factoring by inspection===
It may be possible to express a quadratic equation {{math|''ax''<sup>2</sup> + ''bx'' + ''c'' {{=}} 0}} as a product {{math|(''px'' + ''q'')(''rx'' + ''s'') {{=}} 0}}. In some cases, it is possible, by simple inspection, to determine values of ''p'', ''q'', ''r,'' and ''s'' that make the two forms equivalent to one another. If the quadratic equation is written in the second form, then the "Zero Factor Property" states that the quadratic equation is satisfied if {{math|''px'' + ''q'' {{=}} 0}} or {{math|''rx'' + ''s'' {{=}} 0}}. Solving these two linear equations provides the roots of the quadratic.

For most students, factoring by inspection is the first method of solving quadratic equations to which they are exposed.<ref name=Washington2000>{{cite book|last=Washington|first=Allyn J.|title=Basic Technical Mathematics with Calculus, Seventh Edition|year=2000|publisher=Addison Wesley Longman, Inc.|isbn=978-0-201-35666-3}}</ref>{{rp|202&ndash;207}} If one is given a quadratic equation in the form {{math|''x''<sup>2</sup> + ''bx'' + ''c'' {{=}} 0}}, the sought factorization has the form {{math|(''x'' + ''q'')(''x'' + ''s'')}}, and one has to find two numbers {{math|''q''}} and {{math|''s''}} that add up to {{math| ''b''}} and whose product is {{math|''c''}} (this is sometimes called "Vieta's rule"<ref>{{citation|title=Numbers|series=Graduate Texts in Mathematics|volume=123|first1=Heinz-Dieter|last1=Ebbinghaus|first2=John H.|last2=Ewing|publisher=Springer|year=1991|isbn=9780387974972|page=77|url=https://books.google.com/books?id=OKcKowxXwKkC&pg=PA77}}.</ref> and is related to [[Vieta's formulas]]). As an example, {{math|''x''<sup>2</sup> + 5''x'' + 6}} factors as {{math|(''x'' + 3)(''x'' + 2)}}. The more general case where {{math|''a''}} does not equal {{math|1}} can require a considerable effort in trial and error guess-and-check, assuming that it can be factored at all by inspection.

Except for special cases such as where {{math|''b'' {{=}} 0}} or {{math|''c'' {{=}} 0}}, factoring by inspection only works for quadratic equations that have rational roots. This means that the great majority of quadratic equations that arise in practical applications cannot be solved by factoring by inspection.<ref name=Washington2000/>{{rp|207}}