Turning a described situation into an equation of that shape

What to assume they know

• Standard form and the test for whether an equation is quadratic — The shape an equation has to have before it counts as quadratic
• Expanding a product of two linear expressions in one variable
• Area of a rectangle, and perimeter of a rectangle
• The relation between distance, uniform speed and time
• Forming and solving a linear equation from a worded statement, as met in the earlier classes
• That multiplying both sides of an equation by a number other than zero leaves the same solutions

What they should be able to do

• Choose a single unknown for a described situation and justify why one letter is enough
• Write every other quantity in the situation as an expression in that letter, including quantities obtained by subtracting from a fixed total
• Identify the one sentence in a problem that supplies the equation, and distinguish it from the sentences that only supply definitions
• Explain why a situation involving a product of two variable quantities produces a second-degree term
• Rearrange the resulting equation into standard form, including multiplying through by −1 or clearing a denominator, and say why the solutions are unaffected
• Reproduce the chapter's three modelled situations — the hall, the marbles and the toys — from their stated data alone
• Build equations for the four situations of Exercise 4.1 Question 2, including the one where the unknown appears in a denominator
• State, for each modelled situation, the conditions the answer must satisfy that the equation itself does not record

Where it usually goes wrong

• "Two unknown quantities need two letters." They do not, when a sentence ties them together. The marbles problem gives a total, so the second count is forced; the hall gives a rule, so the length is forced. Introducing a second letter loses the information that made the problem solvable.
• "The equation comes from the question sentence." It does not. The question ("find the length and breadth") tells you what to report; the equation comes from the fact sentence — the stated area, the stated product, the stated total cost.
• "A quadratic turns up because the problem is hard." It turns up because two quantities that both move with x are multiplied together. Name the multiplication and the x² stops being a surprise.
• "Multiplying the whole equation by −1 changes the answer." It does not; the set of numbers satisfying it is untouched. The chapter does this in both parts of Example 1 purely so the x² coefficient comes out positive.
• "Three years from now, only Rohan's age changes." Both ages advance by three. This is the single most common slip in the age item.
• "Lower speed, so less time." A slower train takes longer over the same distance, which is why the 480/(x − 8) term is the larger of the two. Getting the inequality the right way round is what fixes the sign of the 3.
• "Any root of the equation answers the question." The equation knows nothing about lengths being positive or toys coming in whole numbers. Solving is one step; checking admissibility is another.

Questions to check understanding

• "Represent the following situation in the form of a quadratic equation" — the standing instruction of Exercise 4.1 Question 2, where the marks are for the equation and not for solving it
• Area and perimeter items on a rectangle, one dimension described in terms of the other
• Consecutive-integer items, including consecutive even or odd integers
• Age items with a stated difference and a product at a stated time
• Speed–distance–time items in which a change of speed produces a stated change of time, requiring an equation with the unknown in a denominator
• Items asking which root of a formed equation is admissible, and why the other is discarded

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 prayer hall (§4.1, p. 38, with Fig. 4.1). Carpet area 300 square metres; the length is to be one metre more than twice the breadth. Taking the breadth as x metres makes the length (2x + 1) metres. Verified: the area is x(2x + 1) = 2x² + x, so 2x² + x = 300 and the equation is 2x² + x − 300 = 0. Fig. 4.1 shows the rectangle with the two labels and the area written inside.
• Example 1(i), the marbles (§4.2, pp. 39–40). John and Jivanti together hold 45 marbles. Each of them then loses 5. The product of the two remaining counts is 124. Taking John's original count as x makes Jivanti's 45 − x — the chapter marks this step with a "Why?", so make the explanation answer it: the two counts must add to the stated total, so the second is what is left after the first. Verified: after the losses the counts are x − 5 and 40 − x; their product expands to −x² + 45x − 200; setting that equal to 124 and gathering gives −x² + 45x − 324 = 0, and multiplying by −1 gives x² − 45x + 324 = 0.
• Example 1(ii), the toys (§4.2, p. 40). A cottage industry turns out some quantity of toys on a single day. What it costs to make one toy, in rupees, is 55 reduced by that day's output. That day the total production cost was ₹750. Verified: with x toys the unit cost is 55 − x, the total is x(55 − x) = 55x − x², and setting that equal to 750 gives x² − 55x + 750 = 0 after the sign flip.
• Why the square appears — the pattern to draw out across all three. The hall multiplies breadth by a length written in the breadth; the marbles multiply one child's remainder by the other's, both written in x; the toys multiply a count by a price written in that count. In every case two factors both depend on x, so the product contributes an x². Where a problem multiplies a variable quantity by a fixed one, no square appears — worth showing as the contrast case.
• Exercise 4.1 Question 2 (pp. 41–42), four situations to model. Data, then added derivations. (i) A rectangular plot of area 528 m², whose length in metres exceeds double its breadth by one. Verified: with breadth x, the equation is 2x² + x − 528 = 0. (ii) Two consecutive positive integers whose product is 306. Verified: with the smaller as x, x(x + 1) = 306, so x² + x − 306 = 0. (iii) Rohan's mother's age exceeds his own by 26 years, and in three years' time their two ages, in years, will multiply to 360. Verified: with Rohan's present age x, the two future ages are x + 3 and x + 29, their product expands to x² + 32x + 87, and the equation is x² + 32x − 273 = 0. (iv) A train covers 480 km at a uniform speed; had the speed been 8 km/h less, the journey would have taken 3 hours longer. Verified: with speed x km/h, the two times are 480/x and 480/(x − 8), and the second exceeds the first by 3; clearing the denominators gives 3840 = 3x² − 24x, which reduces to x² − 8x − 1280 = 0. This is the only one of the four in which the unknown starts in a denominator, and the only one where the square arrives from clearing fractions rather than from an explicit product.
• The conditions the equations do not carry. The hall's breadth must be a positive length; the marble counts and the toy count must be whole numbers and cannot exceed their totals; Rohan's age must be positive; the train's speed must exceed 8 km/h or the reduced speed is not a speed. The chapter enforces exactly this kind of condition twice later — it discards a negative breadth on p. 44 and a negative distance on p. 46 — so the habit should be established here.

Figures to have open

• Fig. 4.1 (p. 38), the rectangle carrying x, 2x + 1 and its area. Redraw as a schematic; both side labels and the area must be visible together, since the point is that one side is written in terms of the other.
• A two-part bar for the marbles: one bar of 45 split into x and 45 − x, with a slice of 5 removed from each part. Standard schematic; the chapter prints no figure for Example 1.
• A price-against-count sketch for the toys showing the unit cost falling as the count rises, with the bill drawn as a rectangle of area x(55 − x). Standard schematic, and an added device — the chapter states this situation in prose only.
• A two-row journey diagram for the train item: the same 480 km covered twice, at x and at x − 8, with the second row's time bar three hours longer. Standard schematic.

Where this sits in the book

• NCERT Mathematics, Textbook for Class X, Chapter 4 "Quadratic Equations", §4.1 Introduction, p. 38 — the prayer-hall situation and Fig. 4.1
• Same chapter, §4.2 Quadratic Equations, Example 1, pp. 39–40 — the marbles and the toys
• Same chapter, Exercise 4.1 Question 2, pp. 41–42 — the four situations to model
• Forward pointers inside the same chapter: Example 6, p. 44, solves the hall and rejects the negative breadth; Example 8, pp. 45–46, models a distance and rejects a negative root. Both belong to later topics but are the payoff for section 10
• Same chapter, Exercise 4.2 Questions 3–6, p. 44, and Exercise 4.3 Questions 3–5, p. 47, which are further situations of the same kind handled in the later topics
• The chapter's own definition of standard form, §4.2, p. 39