A calculus of the absurd
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4.3.6 The quadratic formula
The quadratic formula 1212 This is the result of completing the square for \(ax^2 + bx + c\) and using it solve the equation \(ax^2 + bx + c = 0\) for \(x\) states that for a quadratic \(ax^2 + bx + c = 0\),
\(\seteqnumber{0}{4.}{50}\)\begin{equation} \label {the quadratic formula} x = \frac {-b \pm \sqrt {b^2 - 4ac}}{2a} \end{equation}
The \(b^2 - 4ac\) gives quite a bit of information away. It’s called the "discriminant", and sometimes written as \(\Delta \).
If \(b^2 - 4ac > 0\) then we have exactly two solutions to the quadratic.
If \(b^2 - 4ac = 0\) (which also means that \(\sqrt {b^2 - 4ac} = 0\)), then there will only be one solution (because adding zero does nothing 1313 This is often handy for algebraic manipulation. - see the equation directly below for details).
\(\seteqnumber{0}{4.}{51}\)\begin{equation*} \frac {-b \pm \sqrt {b^2 - 4ac}}{2a} = \frac {-b}{2a} \text { if $b^2 - 4ac = 0$} \end{equation*}
If \(b^2 - 4ac < 0\) then there are no real solutions to the quadratic.