Skip to content
LearnMathora

Algebra foundations · 04 · The first curve · 9 min

Quadratics

Square the variable and the straight line bends into a parabola — the shape of thrown balls, satellite dishes, and profit curves. Quadratics are your first real curve, and they come with a famous all-purpose key.

Build the intuition

Why x² bends

On a line, each step right adds the same amount. With y = x², each step adds more than the last (1, 3, 5, 7, …) — so the graph curves upward. That accelerating growth is the visual signature of squaring.

y=ax2+bx+cy = ax^2 + bx + c

The vertex is the story's turning point

Every parabola has one extreme point — the bottom of the bowl or the top of the arc. Thrown ball? The vertex is its peak. Profit curve? The vertex is the best price. Finding vertices is why quadratics matter commercially, not just academically.

The quadratic formula, demystified

When ax² + bx + c = 0 won't factor nicely, the formula always delivers both crossing points. It looks intimidating; it's actually just the balance method run once on the general equation, then gift-wrapped for reuse forever.

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

See it move

InteractiveAnatomy of a parabola
0.8
1
-2
y = 0.8(x − 1)² − 2: the lowest point sits at (1, -2), and a sets how gently it bends.

Three dials shape every parabola: slide it, lift it, bend it. The vertex point follows the turning of the story.

A worked example

The thrown ball

  1. A ball is thrown up at 20 m/s from the ground. Physics gives its height:

    h(t)=20t−5t2h(t) = 20t - 5t^2

  2. When does it land? Set h = 0 and factor:

    t(20−5t)=0  ⇒  t=0 or t=4t(20 - 5t) = 0 \;\Rightarrow\; t = 0 \text{ or } t = 4

  3. It lands at t = 4 s. The peak sits midway (symmetry!) at t = 2:

    h(2)=40−20=20 mh(2) = 40 - 20 = 20 \text{ m}

  4. One quadratic answered launch, landing, and peak.

Out in the world

Satellite dishes & headlights

A parabola reflects every incoming parallel ray to a single focus point — and every ray from the focus outward into a parallel beam. Dishes collect with it; headlights project with it. The shape is the function.

Common confusion, cleared

“x² = 9 means x = 3.”

x = −3 works too: squaring eats minus signs. Quadratics typically have two answers, and the ± in the formula is there to hand you both.

“The quadratic formula is for every equation.”

Only for quadratics (highest power 2). Lines don't need it; cubics need more. Match the tool to the degree.

Recap

  • Squaring makes growth accelerate — the graph is a parabola.
  • The vertex is the max or min: the turning point of the story.
  • The quadratic formula reliably finds both roots of ax² + bx + c = 0.