Perfect Squares, Signs and Location of Roots

For $ax^2 + bx + c$:

• it is a perfect square $\iff$ $b^2 - 4ac = 0$ (with $a > 0$);
• it is positive for every $x$ $\iff$ $a > 0$ and $b^2 - 4ac < 0$;
• it is negative for every $x$ $\iff$ $a < 0$ and $b^2 - 4ac < 0$.

Collect the $x$ terms first: $mx^2 + mx + 8x + 9$ has $b = m + 8$.

For real roots of $ax^2 + bx + c = 0$:

• opposite signs $\iff \dfrac ca < 0$;
• both positive $\iff \dfrac ca > 0$ and $-\dfrac ba > 0$; both negative $\iff \dfrac ca > 0$ and $-\dfrac ba < 0$.
• For $a > 0$, $f(k) < 0$ means $k$ lies between the roots. One root in $(p, q)$ and the other in $(q, r)$: $f(p) > 0$, $f(q) < 0$, $f(r) > 0$.
• Integer roots need the discriminant to be a perfect square.