Quadratics and the discriminant

The idea

Completing the square, $a(x + p)^2 + q$, shows the turning point $(-p, q)$. The discriminant $b^2 - 4ac$ counts the real roots: positive gives two, zero gives one repeated root, negative gives none.

1. Rearrange to $ax^2 + bx + c = 0$ and read $a$, $b$ and $c$ with their signs.
2. Use the condition: $b^2 - 4ac > 0$, $= 0$ or $< 0$.
3. Solve the resulting equation or inequality for the unknown constant.

Watch out Real roots means $b^2 - 4ac \ge 0$; two distinct real roots means $> 0$.