Why the sign of b² − 4ac settles how many real roots exist

What to assume they know

• The root formula and how to substitute a, b and c into it — The formula this book hands you for the roots, and how to apply it
• Standard form, and reading a, b and c with their signs — The shape an equation has to have before it counts as quadratic
• Roots by factorising, including the case of a repeated factor — Splitting the middle term, then setting each factor to zero
• That no real number multiplied by itself gives a negative result
• Area and perimeter of a rectangle, and forming an equation from a described situation — Turning a described situation into an equation of that shape

What they should be able to do

• Compute b² − 4ac from an equation in standard form, keeping every sign
• State the three verdicts the sign of b² − 4ac produces, and give the reason for each from the shape of the formula rather than from memory
• Explain why the two roots are always arranged symmetrically about −b⁄2a
• Explain why a negative b² − 4ac leaves no real root, in terms of what squaring a real number can and cannot produce
• Answer a "is this possible?" question by computing a discriminant alone, without solving
• Recognise that a vanishing discriminant and a repeated linear factor are the same situation described two ways, and demonstrate it on a worked example
• Find the value of an unknown coefficient that forces equal roots, and say why one of the algebraic answers may have to be discarded
• Decide whether a described rectangle can exist, using the discriminant of the equation the description produces

Where it usually goes wrong

• "A bigger discriminant means more roots." It means the roots are further apart. Only the sign changes the count.
• "No real roots means no solution, so the question is broken." It means the described situation cannot occur, which is a real and useful answer — Exercise 4.3 Question 4 asks precisely for it.
• "No real roots means there are roots somewhere else." That is true in mathematics beyond this book, but this chapter works entirely within the real numbers and never introduces any other kind; a Class X answer stops at "no real roots".
• "Equal roots means one root." The chapter counts two, and the Summary calls them coincident. The factorising picture explains why: the factor producing that value occurs twice.
• "b² is negative when b is negative." In Example 7, b is −4 and b² is 16. This single slip flips the verdict in that example from none to two.
• "4ac is negative when c is negative." In Example 8, c is −60, so 4ac is −240, and subtracting it adds 240 to the discriminant. Getting this right is what makes 289 rather than −191.
• "If the discriminant is a perfect square the roots are rational." True, but this chapter neither states nor uses it, and a Class X answer should not depend on it. It is offered here so that a student who has met it elsewhere knows where it sits.
• "You still have to solve to know how many roots there are." You do not, and Example 8 is the chapter's demonstration that you do not.

Questions to check understanding

• Settling what kind of roots a given equation has, then producing them where they are real — the task set by Exercise 4.3 Question 1
• Find the value of an unknown coefficient for which a given equation has equal roots, with the non-quadratic case excluded and the exclusion justified
• Find the range of a coefficient for which an equation has two different real roots, or none
• "Is it possible to design…" items on rectangles, groves and parks, where the discriminant is the whole argument
• Items where the answer is that the situation cannot exist, requiring a stated reason rather than a blank
• Prove that a given equation has real roots for every value of a parameter, by showing its discriminant cannot be negative

Examples worth working on the board

Values marked verified are worked out here from the printed data; this chapter prints no answers, and every result below is derived here rather than looked up.

• The split the chapter itself prints (§4.4, p. 44). For a positive b² − 4ac the chapter writes the two roots out separately, each as −b⁄2a with the root term added and then subtracted. Build sections 2 and 3 directly on that printed split: everything else in this topic follows from noticing that only the second term differs between the two roots.
• The three verdicts (§4.4, pp. 44–45). A positive discriminant gives two different real roots; a discriminant of nothing gives two real roots that are the same number, which the chapter obtains by observing that the root term contributes nothing so both roots reduce to −b⁄2a; a negative discriminant gives no real root, because nothing real squares to a negative quantity. The chapter then sets all three out as a numbered list on p. 45, and repeats the list in the Summary on p. 47.
• Why the size does not matter. Verified: the equation x² − 1 = 0 has discriminant 4 and roots 1 and −1; the equation x² − 10000 = 0 has discriminant 40000 and roots 100 and −100. Both have two real roots, and the larger discriminant says only that the roots are further apart, not that there are more of them. Not printed anywhere in the chapter — this is an added contrast, and it is the cheapest way to kill the "bigger discriminant, more roots" error.
• Example 7 (§4.4, p. 45). Equation 2x² − 4x + 3 = 0, so a is 2, b is −4 and c is 3. Verified: b² is 16, 4ac is 24, and the discriminant is −8, which is negative, so no real root exists. One line, no solving. Note that b is negative and b² is positive, which is the sign trap in this example.
• Example 9 (§4.4, p. 46). Equation 3x² − 2x + 1/3 = 0, so a is 3, b is −2 and c is 1/3. Verified: b² is 4 and 4ac is 4, so the discriminant is nothing and the two roots coincide at −b⁄2a, which is 2/6, that is 1/3. Substitution check: 1/3 − 2/3 + 1/3 comes to nothing. This is the chapter's only fully worked zero-discriminant case in §4.4.
• Example 8 (§4.4, pp. 45–46, with Fig. 4.2), which is the argument of this topic in action. A pole must stand on the rim of a circular park 13 m across, its two distances to a pair of gates at opposite ends of a diameter differing by 7 m. The situation produces x² + 7x − 60 = 0. Verified: the discriminant is 49 + 240 = 289, which is positive, so a position exists — and the chapter announces that it exists at this point, before any root is computed. Only afterwards does it locate the pole at 5 m from one gate and 12 m from the other. Section 9 should hold the two halves apart deliberately: existence first, location second.
• The link to factorising (§4.3, p. 43, and Exercise 4.2, p. 44). Verified: Example 5's equation 3x² − 2√6x + 2 = 0 has discriminant 24 − 24, that is nothing — and its factorisation produced the same linear factor twice. The same holds for two items of Exercise 4.2 Question 1: 2x² − x + 1/8 = 0 has discriminant 1 − 1, and 100x² − 20x + 1 = 0 has discriminant 400 − 400, and both factorise as a single linear factor squared. So two of that exercise's five items are zero-discriminant cases, and Example 5 is a third such case living outside the exercise, in §4.3. Verified for contrast: the other three items all have a positive discriminant and two different roots — x² − 3x − 10 = 0 and 2x² + x − 6 = 0 both give 49, and √2x² + 7x + 5√2 = 0 gives 9, its roots being −√2 and −5⁄√2. The chapter never draws this connection; it is added here, and it makes section 10 land.
• Exercise 4.3 Question 2 (p. 47), two equations in which a letter must be chosen so the roots come out equal. For 2x² + kx + 3 = 0, verified: the discriminant is k² − 24, which vanishes at k = 2√6 and at k = −2√6, and both values are usable. For kx(x − 2) + 6 = 0, verified: expanding gives kx² − 2kx + 6 = 0, whose discriminant is 4k² − 24k, that is 4k(k − 6), which vanishes at k = 0 and at k = 6. But k = 0 leaves no x² term at all, so the equation is not quadratic — and what remains, a bare statement that 6 is nothing, is false and has no roots whatever. So only k = 6 answers the question. This item is the chapter's payoff for the a ≠ 0 condition set out in §4.2, and it is the best single question in the exercise.
• Exercise 4.3 Questions 3, 4 and 5 (p. 47), three "is this possible?" items, all answerable by discriminant alone. Question 3: a mango grove, rectangular, its length double the breadth and its area 800 m². Verified: with breadth x the equation is 2x² − 800 = 0, whose discriminant is 6400, positive, so it is possible; the breadth is 20 m and the length 40 m. Question 4: two friends whose ages total 20 years, whose ages four years ago multiplied to 48. Verified: with one age x, the equation is x² − 20x + 112 = 0, whose discriminant is 400 − 448, that is −48, negative — so the situation is impossible and no ages need be reported. This is the only item in the chapter where the honest answer is that the described thing cannot exist. Question 5: a park, rectangular, with 80 m of boundary and an area of 400 m². Verified: with one side x the other is 40 − x, so the equation is x² − 40x + 400 = 0, whose discriminant is 1600 − 1600, nothing — so it is possible, uniquely, and the rectangle is forced to be a 20 m square. The three questions together give one of each case, which is why they belong to this topic rather than to the formula topic.

Figures to have open

• A centre-and-push number line, reusable across sections 4 to 6: −b⁄2a marked fixed, with the two root marks sliding together as the discriminant shrinks and vanishing as it turns negative. This is the central visual of the topic. Standard schematic; the chapter draws nothing of the kind.
• A three-branch decision card for section 7, using the chapter's own wording for the verdicts. Standard schematic.
• Fig. 4.2 (p. 45) recalled in section 9, showing the circle, the gates at opposite ends of a diameter of 13 and the pole on the rim. Redraw as a schematic, adding the right angle at the pole and the two distance labels, neither of which the printed figure carries.
• Three small rectangles for section 11 — the 20 by 40 grove, the impossible pair of ages shown as an empty result, and the 20 by 20 park. Standard schematic.

Where this sits in the book

• NCERT Mathematics, Textbook for Class X, Chapter 4 "Quadratic Equations", §4.4 Nature of Roots, pp. 44–45 — the case-by-case reasoning, the naming of the discriminant, and the three-case list
• Same chapter, §4.4, pp. 45–46 — Example 7, Example 8 with Fig. 4.2, and Example 9
• Same chapter, §4.5 Summary, p. 47, point 5 — the three cases restated, with "coincident" appearing there and not in §4.4
• Same chapter, Exercise 4.3, p. 47 — Questions 1 to 5
• Same chapter, §4.3, p. 43 and Exercise 4.2, p. 44 — the repeated-factor material that section 10 connects to
• Companion topic The formula this book hands you for the roots, and how to apply it owns the formula itself and its derivation; this topic reads the formula rather than establishing it