Describe the solutions to the following quadratic equation:
expr(["+",
["*", A, ["^", "x", 2]],
["*", B, "x"],
C])
= 0
We could use the quadratic formula to solve for the solutions and see what they are, but there's a shortcut...
\qquad
x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}
Substitute the a, b, and
c coefficients from the quadratic
equation:
\qquad\begin{array}
&& b^2-4ac \\ \\
=& (B)^2 - 4 (
A)(C) \\ \\
=& DISCRIMINANT
\end{array}
Because \blue{b^2 - 4ac} = 0, then the
quadratic formula reduces to
\dfrac{-b}{2a}, which means there
is just one rational solution.
Because \blue{b^2 - 4ac} is negative, its
square root is imaginary and the quadratic formula reduces to
\dfrac{-b \pm \sqrt{DISCRIMINANT}}{2a}
, which means there are two complex solutions.
Because \blue{b^2 - 4ac} is a perfect
square, its square root is rational and the
quadratic formula reduces to
\dfrac{-b \pm sqrt(DISCRIMINANT)}{2a}
, which means there are two rational solutions.
Because \blue{b^2 - 4ac} is not a perfect
square, its square root is irrational and the
quadratic formula reduces to
\dfrac{-b \pm \sqrt{DISCRIMINANT}}{2a}
, which means there are two irrational solutions.