Quadratic Equations: quick revision, 15 bites

What Makes an Equation Quadratic?

Degree 2 is the whole game.

Takeaway. If a = 0, the equation collapses to linear — so a ≠ 0 is the defining condition.

The Quadratic Formula: Your Universal Solver

One formula solves every quadratic.

Takeaway. Memorise this formula — it is the fallback when factorisation fails.

The Discriminant Tells You the Nature of Roots

One number predicts your roots.

Takeaway. Compute D first — it tells you what kind of answer to expect before solving.

Solving by Factorisation: Split the Middle Term

Split b, crack the equation open.

Takeaway. Factorisation is quick but only works cleanly when roots are rational.

Completing the Square: Rewrite as a Perfect Square

Turn any quadratic into (x + p)².

Takeaway. Completing the square is the derivation behind the quadratic formula — and the basis for vertex form.

Forgetting the ± When Taking Square Roots

Missing ± costs you half the marks.

Takeaway. Every square root step in a quadratic produces two branches — never drop the negative.

Sign Errors in the Quadratic Formula

One wrong sign ruins both roots.

Takeaway. Substitute into a bracketed template — it prevents the most common sign slips.

Sum and Product of Roots (Vieta's Relations)

Roots hide inside the coefficients.

Takeaway. You can often answer root-related questions without ever solving the equation.

Where Quadratics Show Up in Real Life

Parabolas are everywhere you look.

Takeaway. Any situation with a squared term and a peak or dip is a quadratic in disguise.

The Graph Is a Parabola

Roots are where the curve hits zero.

The discriminant D tells you how many times the parabola crosses the x-axis: twice, once, or never.

Takeaway. Sketching the parabola makes the number and location of roots visually obvious.

Remember the Formula with a Rhythm

Sing it once, never forget it.

Chant: **"minus b, plus or minus, root of b-squared minus four-a-c, all over two-a."** Repeating it aloud three times locks the structure in memory.

Takeaway. Rhythm beats rote — say the formula, don't just read it.

Exam Strategy: Pick the Fastest Method

Right method saves minutes.

Takeaway. Choosing the right method is worth more marks than solving faster.

Quick Check: Verify Your Roots

Two seconds to catch errors.

After finding roots α and β, quickly verify that **α + β = −b/a** and **αβ = c/a**. If either fails, you have made an arithmetic slip.

Takeaway. A 5-second Vieta check catches most calculation mistakes before you submit.

Every Quadratic Has Exactly Two Roots

Two roots — always, counting repeats.

By the **Fundamental Theorem of Algebra**, a quadratic has exactly two roots over the complex numbers. They may be real and distinct, real and equal, or complex conjugates — but there are always two.

Takeaway. If you find only one root, either it is a repeated root or you missed the second.

Building a Quadratic from Its Roots

Reverse the process — roots to equation.

Takeaway. Sum and product of roots is a two-way street — use it to build or to check.