Quadratic equations — factoring, completing the square: Foundations

Before you can follow a single line of the parent note, you need to own every symbol it throws at you. Below, each piece is built from nothing: plain words first, then a picture, then the reason the topic can't live without it. Read top to bottom — each rung of the ladder rests on the one below it.

1. A variable — the letter $x$

Picture it as an empty box $□$ . Writing $x + 2$ just means "whatever number is in the box, add 2 to it." When we later say "solve for $x$ ", we mean: find the number that belongs in the box.

Why the topic needs it: A quadratic equation is a question about an unknown. Without a symbol for "the thing we're hunting", we couldn't even write the hunt down. See 2.1.15-Linear-equations-and-inequalities where solving for one unknown box begins.

2. Powers — what $x^{2}$ means

Picture it literally as a square. If $x$ is the length of one side, then $x^{2}$ is the area of the square with that side.

Figure — Quadratic equations — factoring, completing the square

Why the topic needs it: The word quadratic comes from the Latin quadratus, "square". The whole subject exists because of that $x^{2}$ term. Take it away and you're left with a straight-line (linear) equation — a completely different, easier animal (2.1.15-Linear-equations-and-inequalities).

3. Coefficients and the constant — $a$ , $b$ , $c$

Picture it as three dials $a$ , $b$ , $c$ you set once. After you dial them in, the shape is fixed and only $x$ is free to move.

Why $a \neq = 0$ ? If $a$ were $0$ , the $x^{2}$ term vanishes and we are left with $b x + c = 0$ , a straight-line equation — no longer a quadratic. The condition $a \neq = 0$ is what guarantees the square term survives.

4. The equals-zero sign — why " $= 0$ " is special

Picture it: $A \times B = 0$ is only possible if $A = 0$ or $B = 0$ (or both). You cannot multiply two non-zero numbers and land on zero. Contrast: $A \times B = 12$ tells you almost nothing about $A$ or $B$ individually.

5. Roots — the answers we want

Picture it on a graph. Plotting $y = a x^{2} + b x + c$ for every $x$ draws a smooth U-shaped curve called a parabola (more on that shape in 2.3.4-Parabolas-and-vertex-form). The roots are the exact spots where that curve crosses the horizontal axis — because on that axis the height $y$ is zero.

Why the topic needs it: "Where does the projectile hit the ground?" and "when is profit zero?" both mean find the roots. The parabola picture is the mental home for the whole topic.

6. Factors and factoring — breaking into pieces

Picture it as splitting one wide rectangle of area $x^{2} + 5 x + 6$ into a rectangle whose two side-lengths are $(x + 2)$ and $(x + 3)$ .

Why the topic needs it: Once written as a product $(x + 2) (x + 3) = 0$ , the Zero Product Property (§4) instantly hands you the roots. Factoring converts a hard squared equation into two easy straight-line equations. Deeper factoring tricks live in 3.2.1-Polynomials-factoring-techniques.

7. A perfect square — the shape completing-the-square aims for

Picture it: a big square of side $(x + k)$ . Its area breaks into a small $x \times x$ square ( $x^{2}$ ), two rectangles each $x \times k$ (together $2 k x$ ), and a corner square $k \times k$ ( $k^{2}$ ). Completing the square is nothing but drawing in that missing corner.

Why the topic needs it: A perfect square can be square-rooted directly, and taking a square root is the one operation that lets us undo the $x^{2}$ and free the $x$ . That's the whole trick of Method 2.

8. The square-root sign and $\pm$

Picture it: on the number line, $+ k$ and $- k$ sit at equal distances either side of $0$ . That symmetry is why a parabola has two roots equally spaced around its centre.

Why the topic needs it: Step 6 of the parent's derivation ( $x + \frac{b}{2 a} = \pm \frac{b ^{2} - 4 a c}{2 a}$ ) is exactly where $\pm$ produces the two roots.

9. The discriminant — $b^{2} - 4 a c$

Picture it as a traffic light for the parabola:

positive → the curve crosses the axis at two points (two roots),
zero → the curve just touches the axis (one repeated root),
negative → the curve floats entirely above or below the axis (no real roots — you'd be square-rooting a negative).

This is the gateway to 2.1.17-Quadratic-formuland-discriminant, where each case gets its own full treatment.

How the foundations feed the topic

Factoring (left path) and completing the square (right path) both drain into the same reservoir: solve the quadratic. Different routes, same destination.

Equipment checklist

Test yourself — say the answer aloud before revealing.

What does

x^{2}

mean, and what picture goes with it?

x \times x

; the area of a square of side

x

a x^{2} + b x + c

, which number is the constant term?

c

— the one with no

x

attached.

Why must

a \neq = 0

for a quadratic?

a = 0

the

x^{2}

term disappears and it becomes a linear equation.

State the Zero Product Property.

A \times B = 0

then

A = 0

B = 0

(or both).

What is a root of a quadratic, geometrically?

A value of

x

where the parabola crosses the horizontal axis (height

= 0

To complete the square on

x^{2} + b x

, what do you add?

(\frac{b}{2})^{2}

— half of

b

, then squared.

Why does

k

come with a

\pm

Both

+ k

and

- k

square to

k

, giving two symmetric roots.

What does a negative discriminant

b^{2} - 4 a c

tell you?

No real roots — the parabola never touches the axis.