Chapter 5.5 — Exercise 5.4 — Nature of Roots
Nature of roots of a quadratic equation based on its discriminant. This is Lesson 5 of 5 in Chapter 5: Quadratic Equations.
Knowing Before You Solve
The quadratic formula's ± √(b² − 4ac) hides a genuine shortcut: the single expression under the square root alone tells you how many real roots an equation has, and what kind, before you ever finish solving anything at all. Exercise 5.4 turns that one expression into a real planning tool for every equation still ahead of it.
The Discriminant
b² − 4ac is called the discriminant of the equation ax² + bx + c = 0, and its sign alone fully determines the nature of its roots:
| Discriminant | Nature of roots |
|---|---|
| b² − 4ac > 0 | Two distinct real roots |
| b² − 4ac = 0 | Two equal real roots (a repeated root) |
| b² − 4ac < 0 | No real roots |
This single calculation is far faster than working through the full quadratic formula just to find out whether real solutions exist at all — a genuinely useful first step before committing to solving an equation completely.
This is exactly the same connection between zeroes and graphs covered back in Chapter 3 — a quadratic polynomial's real zeroes are the points where its graph meets the x-axis, and the discriminant is simply the purely algebraic way of asking how many times that meeting happens, without needing to sketch anything by hand at all.
Two Equations, Classified and Solved
For √3x² − 4√3x + 4 = 0 (a=√3, b=−4√3, c=4): b² − 4ac = 48 − 48 = 0, so the roots are equal. Solving confirms it — both roots land on 2/√3. For 2x² − 6x + 3 = 0 (a=2, b=−6, c=3): b² − 4ac = 36 − 24 = 12, positive, so the roots are distinct — working out to (3+√3)/2 and (3−√3)/2.
Notice that the discriminant's sign was checked before reaching for the full formula in both cases — confirming the nature of the roots first makes the final substitution step something to double-check, not something to hope works out. Once the discriminant is known, the ± in the quadratic formula becomes purely mechanical: a positive discriminant guarantees the plus and minus give genuinely different values, a zero discriminant means both signs collapse to the same value, and there was never any suspense about which case applies — the discriminant already settled that beforehand.
Solving for k: Two Equal-Root Problems
Forcing an equation to have equal roots always means setting its discriminant to exactly zero, then solving for whatever unknown constant the equation happens to contain.
- 2x² + kx + 3 = 0: b² − 4ac = 0 gives k² − 24 = 0, so k² = 24 and k = ±2√6.
- kx(x−2) + 6 = 0, k ≠ 0 (which expands to kx² − 2kx + 6 = 0): b² − 4ac = 0 gives 4k² − 24k = 0, which factors to 4k(k−6) = 0, so k = 0 or k = 6 — but k ≠ 0 was given as a condition, so the only valid answer is k = 6.
The second problem is a direct reminder that an algebraic solution can produce a mathematically valid value that the problem's own stated conditions rule out — k = 0 genuinely satisfies b² − 4ac = 0, but it also collapses the original equation back to a non-quadratic one (kx² vanishes entirely when k = 0), which is exactly why the condition k ≠ 0 was attached to the question in the first place. The first problem, by contrast, has no such hidden restriction to worry about — both values of k, +2√6 and −2√6, genuinely produce equal-root quadratics, since the discriminant only ever depends on k² and never on the sign of k itself.
It's worth noticing why a squared unknown like k² = 24 always produces a plus-or-minus pair rather than a single answer: k² = 24 is satisfied by any number whose square is 24, and both +2√6 and −2√6 square to exactly 24. Only extra information outside the discriminant equation itself — like the k ≠ 0 condition in the second problem — can ever narrow a symmetric pair like this down to one specific value.
Can You Even Build It? Three Design Problems
The discriminant answers a question that comes logically before "what are the dimensions" — namely, "is this particular shape even possible at all, given the numbers stated in the problem?"
- A rectangular mango grove, length twice the breadth, area 800 m²: naming the breadth x gives 2x² = 800, or x² − 400 = 0. Here b² − 4ac = 1600 > 0, so real roots exist — the grove is possible, with breadth 20 m and length 40 m.
- Two friends whose ages sum to 20; four years ago their ages multiplied to 48: this leads to x² − 20x + 112 = 0, where b² − 4ac = 400 − 448 = −48, negative. No real roots exist — the situation described is simply impossible, no matter what ages are tried.
- A rectangular park, perimeter 80 m, area 400 m²: this leads to x² − 40x + 400 = 0, where b² − 4ac = 1600 − 1600 = 0 exactly — a repeated root. The park is possible, but only as a very specific shape: both sides equal 20 m, meaning the "rectangle" the problem describes can only actually be built as a square.
These three problems, taken together, cover all three discriminant cases in a single sweep — a design that works cleanly (positive discriminant), one that can't work at all (negative), and one that works only as a single, forced special case (zero) — which is exactly the range of outcomes the discriminant test was introduced to distinguish between. The friends'-ages problem deserves special attention here, since it's the only single place in this entire chapter where the answer to "solve this" is genuinely "there is no solution," not a rejected negative root or an inconvenient fraction, but a mathematical proof that the scenario described could never happen for any real ages at all. Discovering that before spending time hunting for factor pairs or working through the full quadratic formula is precisely what makes the discriminant test worth learning as a first step, not just a footnote to the solving methods covered earlier.
The Chapter, Complete
This exercise closes out Quadratic Equations by adding one diagnostic step in front of every solving method covered earlier — factorisation (Exercise 5.2) and the quadratic formula (Exercise 5.3) both still do the actual solving, but the discriminant now tells you well in advance whether all that solving effort will produce two answers, one, or none at all. For the standard form this entire chapter is built around from the very first page, revisit the chapter introduction. The next chapter, Progressions, moves on from single equations entirely, turning instead to whole sequences of numbers that follow one consistent, predictable pattern.