$$\begin{aligned}x^2 - 4x + 3 &= 0\\(x-3)(x-1) &= 0\\x = 3 \ \text{ or } \ x &= 1\end{aligned}$$

A root = a solution, a value of $x$ that makes the equation true. Quadratics usually have 2 roots.

Reflex: "How many roots?" just means "how many solutions?"

$$\begin{aligned}b^2 - 4ac > 0 &: \ \text{2 real roots}\\b^2 - 4ac = 0 &: \ \text{1 real root}\\b^2 - 4ac < 0 &: \ \text{no real roots}\end{aligned}$$

'Real' just means a real number — one real root is just one root.

Reflex: Check the sign of $b^2-4ac$ before finishing the quadratic formula.

This is for intersection questions: how many times does a line meet a curve?

Sub the line's equation into the curve's, then arrange everything on one side into $ax^2 + bx + c = 0$. Every root of that quadratic is one meeting point — so $b^2 - 4ac$ counts the intersections:

$$\begin{aligned}b^2 - 4ac > 0 &: \ \text{cuts twice}\\b^2 - 4ac = 0 &: \ \text{touches once (tangent)}\\b^2 - 4ac < 0 &: \ \text{misses completely}\end{aligned}$$

Reflex: "How many times do they meet?" — sub in, arrange, then $b^2-4ac$. You never need to solve for $x$.

$$\begin{aligned}\text{always } + &: \ a > 0, \ \ b^2 - 4ac < 0\\\text{always } - &: \ a < 0, \ \ b^2 - 4ac < 0\end{aligned}$$

Two conditions, both needed: the curve opens the right way (that is the sign of $a$), and it never crosses the $x$-axis (that is $b^2-4ac<0$). Write "opens upwards" in your answer — "smiley face" helps you remember it, but it is not accepted in formal marking.

Reflex: Never stop at $b^2-4ac<0$ — check the sign of $a$ to reject one of the two ranges.

$$b^2 - 4ac = \underbrace{A(k+h)^2}_{\geq\,0} + C$$

Compute $b^2-4ac$, expand, simplify; complete the square if the sign isn't obvious — after that the sign is visible at a glance.

Reflex: Prove the sign for every parameter value — never just sub into $b^2-4ac=0$.

Exam questions hide the condition in words. Translate the phrase into a $b^2-4ac$ condition first, then do the algebra:

$$\begin{aligned}\text{“two distinct roots”} &: \ b^2-4ac > 0\\\text{“equal roots” / “tangent”} &: \ b^2-4ac = 0\\\text{“real roots”} &: \ b^2-4ac \geq 0\\\text{“no real roots”} &: \ b^2-4ac < 0\end{aligned}$$

Graph language translates the same way: "cuts twice" → distinct, "touches once" → equal (tangent), "does not meet" → no real roots. And "real roots" allows 1 or 2 of them — that is why it gets $\geq$, not $>$.

Reflex: Translate the words first — after that, every question is the same four conditions.