Quadratic formula — derivation by completing the square: Foundations

0. What is a variable, really?

Think of $x$ as a sealed box. We don't know what's inside, but the equation is a list of clues about it. Solving the equation means opening the box.

1. Powers and the symbol $x^{2}$

The word "square" is not an accident — it is a picture.

Look at the figure: a square whose side is $x$ has area $x \times x = x^{2}$ . That is why we call it "squaring", and it is the seed of the whole method — later we literally rearrange rectangles to complete a square.

Recall Why is squaring the hard part?

Because the same box $x$ shows up squared and on its own in $x^{2} + \dots x$ . We cannot just read the answer off — squaring hides it. The entire method exists to tame that squared term. ::: Because the unknown appears both squared and linear, so it can't be read off directly.

2. Coefficients: $a$ , $b$ , $c$

Recall Why must

$a \neq = 0$ ? The term $a x^{2}$ is the only place $x^{2}$ appears. If $a = 0$ that square vanishes and there is no square left to complete — the equation collapses to a straight-line (linear) one. ::: Because $a = 0$ removes the $x^{2}$ term entirely, so it is no longer quadratic.

3. The equals sign and "= 0"

The red dots in the figure are the crossings — the roots. The three curves show the only three things that can happen: a parabola can cross the ground twice (cyan), touch it once (amber), or miss it entirely (white). That three-way split is exactly what the discriminant (Section 8) will predict.

4. Doing the same to both sides

5. Fractions and a common denominator

Dividing by $a$ produces fractions like $\frac{b}{a}$ and $\frac{c}{a}$ , so we must be fluent with them.

6. Perfect squares and the identity $(x + h)^{2}$

Read this identity right-to-left and it becomes a factory: given $x^{2} + (linear)$ , what $h$ turns it into a clean square?

The figure shows why: a big square of side $(x + h)$ splits into an $x^{2}$ block, two identical $h \cdot x$ strips (that's the $2 h x$ ), and a tiny $h^{2}$ corner. To "complete the square" we supply exactly that missing corner $h^{2}$ .

See Perfect square trinomials and Completing the square technique for the full drill.

7. Absolute value, square roots, and the symbol $\pm$

Recall Why

$4 a^{2} = 2∣ a ∣$ , not $2 a$ ? Because a square root is never negative, and $4 a^{2} = 2∣ a ∣$ where $∣ a ∣$ is the absolute value from above. But since the $\pm$ out front already supplies both signs, writing $2 a$ instead of $2∣ a ∣$ changes nothing about the pair of answers. ::: The $\pm$ absorbs the sign, so $2 a$ and $2∣ a ∣$ give the same two roots.

2

x^{2}

tell you to do?

Multiply

x

by itself once:

x \times x

a x^{2} + b x + c

, which letter is the constant, and is it always positive?

c

is the constant; it can be negative (its sign is part of

c

Why is

a \neq = 0

required?

a = 0

the

x^{2}

term disappears and it is no longer quadratic.

What single rule lets us divide the whole equation by

a

You may do the same operation to both sides of a balanced equation (allowed because

a \neq = 0

To make

x^{2} + k x

a perfect square, what do you add?

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

Expand

(x + h)^{2}

x^{2} + 2 h x + h^{2}

What does

∣ - 3 ∣

equal, and why?

3

— absolute value is distance from zero, always positive.

Why does taking a square root introduce

\pm

Because both a positive and a negative number square to the same value, so

y^{2} = k

has two solutions.

What is the discriminant and where does it live in the formula?

Δ = b^{2} - 4 a c

; it sits inside the square root.

What does

Δ = 0

predict geometrically?

The parabola just touches the ground line — one repeated root.

To add

- \frac{c}{a} + \frac{b ^{2}}{4 a ^{2}}

, what must you do first?

Give them a common denominator (

4 a^{2}

), then add the numerators.

Foundations — Quadratic formula — derivation by completing the square