Foundations — Formation of quadratic with given roots

This page assumes you know nothing. Before we can talk about "forming a quadratic from given roots" over on the parent note, we must earn every symbol it silently uses. We go one brick at a time — each brick sits on the last.

Brick 0 — The number line and the letter $x$

Look at the figure. The ruler runs left (negative) to right (positive), with $0$ in the middle. The little pointer sitting on the line is the current value of $x$. When we later "plug in $x=3$", we are just sliding that pointer to the tick labelled $3$.

Why does the topic need this? Because a quadratic is a machine that eats an $x$ and spits out a number. Without a movable input there is nothing to feed the machine.

Brick 1 — A function and the notation $P(x)$

The bracket does not mean "P times x". It means "P applied to x". These are wildly different:

Why the topic needs it: the whole idea "$\alpha$ is a root of $P(x)$" is a statement about what the machine outputs, so we need a name for that output.

Brick 2 — Exponents and the word "quadratic"

• $x^1=x$ (one copy)
• $x^2=x\times x$ (two copies)
• $x^0=1$ (zero copies — an agreed convention)

Why "$a\ne 0$"? If $a=0$ the highest power drops to $1$ and the curve becomes a straight line, not the bowl-shape we care about.

Brick 3 — The parabola: what a quadratic looks like

In the figure the horizontal ruler is the x-axis (inputs), the vertical ruler is the y-axis (outputs = heights). The curve dips below the axis, then rises back up — a bowl. If $a>0$ the bowl opens upward (a smile); if $a<0$ it opens downward (a frown). We will need both cases later, so keep them in mind.

Why the topic needs it: "roots" will turn out to be where this bowl touches the x-axis, so we must first be able to see the bowl.

Brick 4 — Zero, and what "crossing the x-axis" means

The x-axis is the horizontal line where the height is $0$. A point sits on the x-axis exactly when its height $P(x)$ equals $0$.

This single idea — "height $=0$" — is the doorway to everything that follows.

Brick 5 — Greek letters $\alpha$, $\beta$ and the word "root"

In the figure the two floor-touch points are marked $\alpha$ and $\beta$. Slide the pointer to $x=\alpha$: the curve is exactly on the floor, so $P(\alpha )=0$. Same at $x=\beta$.

Brick 6 — The factor $(x-\alpha )$ and why the sign flips

Here is the star fact of the whole topic, straight from the Factor theorem:

Why $(x-\alpha )$ and not $(x+\alpha )$? Because we want a piece that becomes zero at $x=\alpha$. Substitute: $(x-\alpha ){|}_{x=\alpha }=\alpha -\alpha =0.✓$ If instead we wrote $(x+\alpha )$, plugging in $x=\alpha$ gives $2\alpha$, which is not zero (unless $\alpha =0$). So the minus sign is forced — it is the only choice that switches off at the right spot.

Brick 7 — Multiplying two factors: "switch off if either is off"

Why do we multiply the two factors $(x-\alpha )$ and $(x-\beta )$ instead of adding them?

$P(x)=(x-\alpha )(x-\beta )$

• At $x=\alpha$: first bracket $=0$, whole product $=0$. ✔
• At $x=\beta$: second bracket $=0$, whole product $=0$. ✔
• Anywhere else: both brackets non-zero, so the product is non-zero — the curve is off the floor. ✔

This is the engine of the parent note. Multiplication (not addition) is the tool that says "zero if either root is hit".

Brick 8 — The stretch constant $k$

$P(x)=k(x-\alpha )(x-\beta ),k\ne 0$

All three curves in the figure hit the floor at the same two points, yet a bigger $|k|$ pinches the bowl narrower and a smaller $|k|$ flattens it. A negative $k$ flips the smile into a frown. See Graph transformations for the general story of vertical stretching.

Why the topic needs it: roots fix where the curve lands but not how stretched it is — so there is a whole family of quadratics with the same roots, and $k$ is the dial that separates them.

Brick 9 — Expanding brackets (turning factor-form into $a{x}^2+bx+c$)

To reach the familiar $a{x}^2+bx+c$ we expand — multiply everything out. Multiply each term of the first bracket by each term of the second:

$(x-\alpha )(x-\beta )=x\cdot x+x\cdot (-\beta )+(-\alpha )\cdot x+(-\alpha )(-\beta )$ $=x^2-\beta x-\alpha x+\alpha \beta =x^2-(\alpha +\beta )x+\alpha \beta .$

Notice two combined pieces:

• $\alpha +\beta$ — the sum of the roots, which appears with a minus sign.
• $\alpha \beta$ — the product of the roots, the plain constant at the end.

These two combinations are the heart of Vieta's formulas, and they are why the parent note can build a quadratic from sum and product alone: $x^2-(\text{sum})x+(\text{product})=0.$

Brick 10 — The coefficients $a$, $b$, $c$

From these you can read off Vieta's relations $\alpha +\beta =-\frac{b}{a}$ and $\alpha \beta =\frac{c}{a}$ — but that is the parent note's job. Your job here was only to make sure not one of $x$, $P(x)$, $x^2$, $\alpha$, root, factor, $k$, $a$, $b$, $c$ was ever a mystery.

How the bricks feed the topic

Equipment checklist

Test yourself — say the answer out loud, then reveal.