This page assumes nothing. If the parent note used a letter, a squiggle, or a word without explaining it, we build it here from the ground up, in an order where each idea rests on the one before it.
The picture: imagine a number line. x is a slider that can sit anywhere on it. A coefficient like 3 stretches whatever x is by a factor of three.
Why the topic needs it: in §3 we will meet an equation with three coefficients standing side by side. Everything the topic decides is built out of those three known numbers — so first you must be able to read "the number attached to a letter".
Before we can talk about a graph, we need the letter y.
Three separate curves are drawn above:
blue dips below the axis → it must cross twice → two real roots.
yellow just kisses the axis at its vertex → one repeated root.
pink floats entirely above the axis → it never crosses → no real roots at all.
That pink curve is a puzzle: an equation with no real answer. Rather than call it "impossible", mathematicians later invented a new kind of number to give it answers. We build exactly that in §7 — for now, just notice the pink curve exists and hold the question in mind.
Why the topic needs it: "nature of roots" is literally the question which of these three pictures am I looking at? — and one number will answer it without drawing anything.
See also Parabola and Axis of Symmetry and Graphing Quadratic Functions.
We now earn the discriminant instead of borrowing it. The one tool we need is Completing the Square: rewriting the equation so the unknown appears only inside a single squared bracket, because a square is the one shape we know how to un-do (with and ± from §5). Watch where the pieces go.
Step 1 — divide by a so the curving coefficient is 1.x2+abx+ac=0Why: completing the square only works cleanly when x2 stands alone. (We may divide because a=0, §3.)
Step 2 — build the perfect square. The bracket (x+2ab)2 expands to x2+abx+4a2b2. It has the two terms we want plus a spare 4a2b2, so we add that spare and immediately subtract it back:
(x+2ab)2−4a2b2+ac=0What it looks like: the whole tilted curve has been re-centred so the mirror line sits at x=−2ab — the axis of symmetry of §4 appearing on its own.
Step 3 — move the leftovers to the right and combine into one fraction.(x+2ab)2=4a2b2−ac=4a2b2−4acWhy: the constants 4a2b2 and ac share the denominator 4a2 (multiply the second top and bottom by 4a), and their numerators fuse into the single expression b2−4ac. This is the exact moment b2−4ac is born — it is what is left over after the square is completed.
Step 4 — un-do the square (this is why §5 mattered).x+2ab=±4a2b2−4ac=2a±b2−4ac
Step 5 — isolate x.x=2a−b±b2−4ac
Read the result as centre plus-or-minus a step, both carrying the same divisor 2a that Step 4 handed them:
2a−b — this fraction is the axis of symmetry (§4); it is where the vertex sits.
2a±b2−4ac — the sideways step out to each root, whose size is fixed by the number under the root.
Δ>0: real step out → land at two spots (two real roots).
Δ=0: step size zero → both spots collapse to the vertex (one repeated root).
Δ<0: root of a negative → the step leaves the real line (complex roots — §7).
The figure shows the same axis of symmetry −2ab with three step-sizes: wide (two roots), zero (equal roots), and imaginary (complex roots leaving the line).
Read it as a flow: numbers-with-letters and the squaring-fold build the standard form; that draws a parabola whose axis crossings are the roots; the square root and completing the square assemble the formula; the discriminant sits inside it and, together with complex numbers, decides the final nature of the roots.
