Chapter 5.1 — Introduction to Quadratic Equations

The Standard Form

Any equation p(x) = 0, where p(x) is a polynomial of degree 2, is a quadratic equation. Writing its terms in descending order of degree gives the standard form:

Identifying a, b, and c correctly — including when one of them is zero — is the first skill this chapter depends on:

The middle two rows are the ones worth studying closely: whenever a term is simply absent from how an equation is written, its coefficient is 0, not some value to guess at — 5y² − 1 = 0 has no y-term at all, so b = 0 exactly, and x² − x = 0 has no plain number at the end, so c = 0 exactly. The condition a ≠ 0 is what actually makes an equation quadratic in the first place — if a were 0, the x² term would vanish and the equation would collapse back to linear, which is precisely why every definition of "quadratic" leads with that one condition.

Quadratic equations aren't just an algebra exercise — they show up directly whenever a problem involves multiplying two related quantities together, which is exactly why areas, speeds, and profit calculations lead to them so often. A rectangle's area is length times width; if one of those dimensions is described in terms of the other ("3 more than twice the width"), multiplying them together produces an x² term automatically, without anyone setting out to write a quadratic equation on purpose.

Six Equations, Tested

Deciding whether an equation is genuinely quadratic means simplifying it fully first, then checking the highest power of x that survives — not just glancing at how it's written.

• x² − 6x − 4 = 0 — already in standard form with the highest power visibly 2. Quadratic, and no further work is needed to confirm it.
• x³ − 6x² + 2x − 1 = 0 — the x³ term is the highest power present, giving degree 3 overall. Not quadratic, despite also containing an x² term — a single higher-power term anywhere in the equation is enough to disqualify the whole thing, regardless of what else is present.
• 7x = 2x² — rearranges to 2x² − 7x = 0, degree 2. Quadratic, even though it wasn't written with a constant term at all — c = 0 here, exactly as in the x² − x = 0 example above.
• x² + 1/x² = 2 (x ≠ 0) — multiplying through by x² to clear the fraction gives x⁴ + 1 = 2x², degree 4. Not quadratic — the fraction hides a higher degree that only appears once it's cleared, meaning judging this equation by its original written form alone would give the wrong answer.
• (2x+1)/(3x+1) = (bx−1)/(x−2) — cross-multiplying and simplifying gives (6−b)x² + (5+3b)x + (1−2b) = 0. Quadratic for any b ≠ 6 — if b were exactly 6, the coefficient of x² would become 6−6=0, cancelling the squared term entirely and dropping the equation back to linear.
• 3y² = 192 — degree 2 as written, using y instead of x purely as a labelling choice. Quadratic — the letter used for the variable never has any bearing on whether an equation counts as quadratic.

The fourth and fifth examples share a lesson worth remembering: an equation's true degree can be hidden behind fractions, and only shows itself once those fractions are cleared by cross-multiplying or multiplying through by the denominator. Skipping that clearing step and judging degree from the equation's original, unsimplified form is the single most common way to misclassify an equation in this chapter.

What "Solving" a Quadratic Will Mean

A specific value of x that makes ax² + bx + c = 0 true is called a root of the equation — the same idea as a zero of the corresponding polynomial, covered back in Chapter 3. Because a quadratic polynomial can have at most two zeroes, every quadratic equation has at most two roots, never more. Some quadratics have two distinct roots, some have a single repeated root, and — as later exercises in this chapter show in detail — some have no real roots at all. Finding out which case applies to a given equation, and then finding the actual root or roots themselves, is what the rest of this chapter is about.

Unlike the linear equations from the previous chapter, a quadratic can't generally be solved by simply isolating x on one side — the x² term makes that impossible in one step, since x appears in two different forms (x and x²) that can't be combined into a single term through ordinary rearrangement. This chapter builds up two genuinely different techniques for getting around that: factorisation, which works by splitting the equation into two simpler linear pieces, and a more general algebraic method — completing the square — that works even when no clean factorisation exists, no matter how awkward the coefficients look.

It's worth previewing why two techniques are needed at all, rather than one. Factorisation is fast whenever a quadratic happens to split into two factors with whole-number or simple-fraction coefficients — but plenty of genuine, real-world quadratics simply don't split that cleanly, and no amount of patient searching for the right factor pair will ever succeed in finding one that simply doesn't exist. Completing the square sidesteps that problem entirely by working algebraically rather than by search, which is exactly why it succeeds on every quadratic equation without exception, and why it's the method used to derive the general-purpose quadratic formula later in the chapter.

Where This Chapter Goes Next

This introduction covers only the definition — recognising a quadratic equation and correctly identifying its coefficients a, b, and c, including whenever one of them happens to be zero. Exercise 5.1 practices that recognition further on trickier equations, then builds fresh quadratic equations directly from word problems; Exercise 5.2 introduces the first solving method, factorisation, and applies it to a wide range of word problems; Exercise 5.3 derives the general quadratic formula from scratch and puts it to work on equations factorisation alone can't easily crack; and Exercise 5.4 closes the chapter by showing how to tell what kind of roots an equation has — two, one, or none — before even attempting to solve it. For the linear-equation groundwork this chapter builds on, see Chapter 4's introduction, where the same kind of standard-form thinking was first introduced for degree-1 equations.