What does it mean if the discriminant of a discriminant is negative? For example $x^2+(m+1)x+(m-2)=0$
$b^2-4ac=m^2-2m+9$ and the discriminant of that $=-32$
Or does this just not work? Thanks in advance!
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$\begingroup$What this means is that $m^2-2m+9$ has no real roots, i.e., it is always either positive or negative (because polynomials are continuous, there are no "gaps", so for the function to switch sign it would need to cross the x-axis and would have a root). To find out which, we evaluate at $m=1$:
$$1^2-1 \cdot 2 + 9 =1-2+9=8$$
This is positive, and therefore we can say that $m^2-2m+9$ is always positive.
This means $b^2-4ac$ of the original function is always positive. This tells you that your original function, no matter what the value of $m$ is, has two real roots.
In general, if the discriminant of a function $f(x)$ is a function $g(m)$, if the discriminant of $g(m)$ is always negative, then $f(x)$'s discriminant ($g(m)$) is either always negative for any value of $m$ or always positive. You can determine which by evaluating $g(0)$ or any similarly easy evaluation. If the discriminant of $f(x)$ is always negative for any value of $m$, it means that $f$ is guaranteed to have no real roots. If the discriminant of $f(x)$ is always positive, it means that $f$ is guaranteed to have two real roots.
On the opposite side, if $g(m)$'s discriminant is positive, then the number of real roots of $f$ varies depending on the value of $m$ chosen.
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