Why a root of the equation is the same thing as a zero of the polynomial

What to assume they know

• Standard form and the test for a quadratic equation — The shape an equation has to have before it counts as quadratic
• Zeroes of a polynomial, and the result from Class X Chapter 2 that a quadratic polynomial has at most two of them
• Substituting a number for the variable in an expression and evaluating it, including with fractions and surds
• The distinction between an expression, which has a value, and an equation, which is a claim that can be true or false
• Arithmetic with surds: squaring √2 and √3, and multiplying √6 by √2

What they should be able to do

• State what it takes for a number to be a root of a quadratic equation, in the general form the chapter gives with α
• Test a proposed number by substitution and report whether it is a root, showing the evaluation rather than asserting the verdict
• Explain why the numbers at which ax² + bx + c evaluates to nothing are exactly the numbers that solve the equation formed by setting it to zero
• Use the three words the chapter uses — root, solution, satisfies — correctly and interchangeably, and say which object each is naturally attached to
• Deduce the "at most two roots" ceiling from the Chapter 2 result about zeroes, rather than stating it as a separate rule
• Explain the sense in which an equation can be said to have two roots when only one distinct number solves it, using the chapter's own repeated-factor example
• Distinguish the cost of checking a proposed root from the cost of finding one, and say why the chapter asks the reader to verify at the end of two worked examples

Where it usually goes wrong

• "A polynomial and an equation are different things, so their answers are different too." They are different things — one is an expression, the other a claim — but the numbers picked out are identical, because both are picked out by the same test.
• "A root is something you get by solving." A root is a number with a property. Solving is one way to find such numbers; substituting is the way to confirm one. The chapter's opening does the second without doing the first.
• "At most two roots is a rule about quadratics you have to remember." It is inherited from the zeroes result of Chapter 2. Students who see the inheritance do not have to remember it separately.
• "If only one number works, the equation has one root." The chapter's own Example 5 reports the same value twice, because the factor producing it occurs twice. Both descriptions are in play, and §4.5 later calls this case two coincident roots.
• "Verifying is optional tidying." It is the only step in this chapter that can be done without trusting a method, and it is the only defence against a sign slip in the factorising.
• "Checking a root and finding a root are equally hard." Checking is one substitution and needs nothing but arithmetic. Finding needs a method, which is what the rest of the chapter supplies.
• "α means something new." It is a letter standing for a particular number, used because x is already doing the job of the variable. The chapter changes letter to keep the two roles apart.

Questions to check understanding

• "Show that the given value is a root of the given equation" — a pure substitution item, common as a one- or two-mark question
• Given that a stated number is a root, find the value of an unknown coefficient
• State the maximum number of roots a quadratic equation can have, with a reason
• Items that supply a factorised equation and ask for its roots, testing the identification directly
• Verification items appended to a solving question, where the marks are for the substitution and not for the answer
• Short-answer items asking for the difference in meaning between a zero of a polynomial and a root of an equation, where the expected answer is that there is none in the numbers picked out

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 chapter's opening test (§4.3, p. 42). The equation is 2x² − 3x + 1 = 0 and the candidate is 1. Verified: the terms come to 2, then −3, then 1, and they total nothing, so 1 passes. The chapter then makes the pivot the whole topic rests on: the same computation shows 1 is a zero of the polynomial 2x² − 3x + 1. Show the substitution once and label the identical result twice.
• The general statement (§4.3, p. 42). For a real number α and a quadratic equation in standard form, α earns the name root exactly when replacing the variable by α makes the expression evaluate to nothing. The chapter uses α here and again in the Summary on p. 47, point 2; use the same letter so a student reading the book recognises it.
• A candidate that fails, for contrast. Not printed, but needed: test 2 in 2x² − 3x + 1. Verified: 8 − 6 + 1 = 3, which is not nothing, so 2 is not a root. Every worked example in the chapter tests numbers that pass; an explanation that never shows a failure teaches substitution as ceremony.
• The three roots the chapter finds by factorising, offered here as substitution exercises rather than as results to be taken on trust. From Example 3 (p. 42), the equation 2x² − 5x + 3 = 0 with candidates 1 and 3/2. Verified: at 1 the terms are 2, −5, 3, totalling nothing; at 3/2 they are 9/2, −15/2, 3, also totalling nothing. From Example 4 (p. 43), the equation 6x² − x − 2 = 0 with candidates 2/3 and −1/2. Verified: at 2/3 the terms are 8/3, −2/3, −2, totalling nothing; at −1/2 they are 3/2, 1/2, −2, totalling nothing. The chapter says outright that it verifies these two. From Example 5 (p. 43), the equation 3x² − 2√6x + 2 = 0 with candidate √2⁄√3. Verified: the first term is 3 × 2/3 = 2, the middle term is −2√6 × √2⁄√3 = −2√4 = −4, the last is 2, and 2 − 4 + 2 is nothing. This is the surd check worth doing slowly.
• Example 5's repetition (§4.3, p. 43). The factorisation produces the same linear factor twice, and the chapter reports the root twice as a result. So there is one distinct number that solves the equation and two roots in the chapter's counting. Verified: the two reported roots are the same value, √2⁄√3. Section 8 should hold both statements at once rather than choosing between them.
• The ceiling, and its source. The chapter cites Chapter 2 for the fact that a quadratic polynomial has at most two zeroes, and then transfers it. Zeroes of the polynomial and roots of the equation are the same numbers; the polynomial has at most two zeroes; therefore the equation has at most two roots. No new argument is made at this step, and saying so is the point.
• The Summary restatement (§4.5, p. 47, point 2) repeats the α definition and the identification of zeroes with roots. Useful as the closing card of the explanation, because it shows the chapter itself treats this as one of its five keepable results.

Figures to have open

• A substitution card: the polynomial written once, a value dropped into every occurrence of the variable, and the running total. Standard schematic; the chapter prints no figure in §4.3.
• A two-panel identity card showing the expression on one side and the equation on the other, with a single test arrow feeding both. Standard schematic, and an added device.
• A number-line strip carrying the roots found in Examples 3, 4 and 5, with the repeated one marked as a single point that answers twice. Standard schematic; the chapter draws nothing of the kind.
• No printed figure is needed for this topic. Fig. 4.1 and Fig. 4.2 both belong elsewhere in the chapter.

Where this sits in the book

• NCERT Mathematics, Textbook for Class X, Chapter 4 "Quadratic Equations", §4.3 Solution of a Quadratic Equation by Factorisation, p. 42 — the opening paragraph, the α statement, and the sentence transferring the Chapter 2 ceiling
• Same chapter, Examples 3, 4 and 5, pp. 42–43 — used here as substitution material; their factorising belongs to Splitting the middle term, then setting each factor to zero
• Same chapter, §4.5 Summary, p. 47, point 2 — the restatement
• Backward pointer: Class X Chapter 2, for zeroes of a quadratic polynomial and for the at-most-two result this topic borrows
• Forward pointer inside the same chapter: §4.4, pp. 44–45, where the two-equal-roots case is named and connected to the discriminant