The shape an equation has to have before it counts as quadratic

What to assume they know

Polynomials in one variable, their degree, and the quadratic polynomial written in the form ax² + bx + c — Class X Chapter 2
Expanding a product of two binomials, and the expansions of (x + k)² and (x + k)³
Collecting like terms and moving terms across an equals sign
What a real number is, and that real numbers include integers, fractions and surds such as √2
The difference between an expression and an equation

What they should be able to do

State the four conditions the standard form imposes: one variable, real coefficients, highest power exactly two, and a leading coefficient that is not zero
Explain why allowing the leading coefficient to be zero would destroy the definition, and say what the equation becomes instead
Recognise a quadratic equation written with its terms out of order, and rewrite it in standard form by arranging the powers downwards
Given an equation with brackets or powers on both sides, expand, collect and cancel, and then decide whether what remains is quadratic
Identify a, b and c, with their signs, from an equation already in standard form
Explain why an equation whose two sides both carry x² may still fail to be quadratic, using item (ii) of the chapter's Example 2
Explain why an equation whose two sides both carry x³ may nevertheless be quadratic, using item (iv) of that same Example 2
Sort the eight items of Exercise 4.1 Question 1 into quadratic and not, giving the simplified equation as the evidence in each case

Where it usually goes wrong

"It has an x², so it is quadratic." Item (ii) of Example 2 has an x² on both sides and is a linear equation. The x² has to survive the tidying.
"It has an x³, so it cannot be quadratic." Item (iv) has cubes on both sides and they cancel exactly. Degree is what is left, not what was written.
"Standard form is just neatness." It is the form in which a, b and c can be read off at all, and every later tool in the chapter — factorising, the discriminant, the formula — takes a, b and c as its input.
"a can be any number." If a is zero the x² term is gone and the definition no longer applies; what remains is a linear equation. This is not a technicality — Exercise 4.3 Question 2(ii) turns on exactly this point.
"b or c must be present." They need not be. An equation such as −x² + 301 = 0 has no x term at all and is still quadratic.
"The coefficients have to be whole numbers." The definition asks only that they be real, and Exercise 4.2 supplies items with √2 and with ⅛ as coefficients.
"Dividing the whole equation by 6 changes the problem." Multiplying or dividing an equation by a number that is not zero leaves exactly the same set of solutions, which is why the chapter is free to reduce 6x² + 12x + 12 = 0.

Questions to check understanding

Sorting a list of eight given equations into quadratic and not — the task of Exercise 4.1 Question 1, with the simplified equation demanded as justification
Rewrite a given equation in standard form and state a, b and c with signs
Given an equation with a letter in the coefficient of x², state the values of that letter for which the equation is quadratic
Find the value of a constant for which a given equation reduces to a linear one
One-mark identification items asking only for the degree of the tidied equation
Justification items: explain why an equation carrying x³ on both sides can still be quadratic

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). A charity trust wants the carpet to cover 300 square metres, and wants the length to exceed twice the breadth by one metre. Taking the breadth as x metres makes the length (2x + 1) metres. Verified: the area is x(2x + 1) = 2x² + x, so the condition reads 2x² + x = 300, and moving 300 across gives 2x² + x − 300 = 0.
Fig. 4.1 itself (p. 38). A single rectangle with a patterned fill, the short side marked x, the long side marked 2x + 1, and 300 m² set inside it. Nothing else is drawn — no scale, no units on the sides.
Four equations the chapter calls quadratic (§4.2, p. 39), useful because two of the four arrive already tidy and two do not: 2x² + x − 300 = 0; 2x² − 3x + 1 = 0; 4x − 3x² + 2 = 0; and 1 − x² + 300 = 0. Verified: the third rearranges to −3x² + 4x + 2 = 0, and the fourth collapses to −x² + 301 = 0, which is quadratic with b equal to zero. Use the fourth to make the point that b and c may vanish and a may not.
Example 2 (§4.2, pp. 40–41), four items, and the chapter's Remark on p. 41 singles out (ii) and (iv) as the two that mislead. (i) (x − 2)² + 1 = 2x − 3. Verified: the left side expands to x² − 4x + 5, so the equation becomes x² − 6x + 8 = 0 — quadratic, with a = 1, b = −6, c = 8. (ii) x(x + 1) + 8 = (x + 2)(x − 2). Verified: left is x² + x + 8, right is x² − 4, the squares cancel, and what is left is x + 12 = 0 — degree one, so not quadratic. (iii) x(2x + 3) = x² + 1. Verified: 2x² + 3x = x² + 1 leaves x² + 3x − 1 = 0 — quadratic. (iv) (x + 2)³ = x³ − 4. Verified: the left side expands to x³ + 6x² + 12x + 8, the cubes cancel, and 6x² + 12x + 12 = 0 remains, which divides through by 6 to x² + 2x + 2 = 0 — quadratic, despite the cube on both sides.
Exercise 4.1 Question 1 (p. 41), eight items to sort. (ii) x² − 2x = (−2)(3 − x) → verified x² − 4x + 6 = 0, quadratic. (iii) (x − 2)(x + 1) = (x − 1)(x + 3) → verified −3x + 1 = 0, not quadratic. (iv) (x − 3)(2x + 1) = x(x + 5) → verified x² − 10x − 3 = 0, quadratic. (v) (2x − 1)(x − 3) = (x + 5)(x − 1) → verified x² − 11x + 8 = 0, quadratic. (vi) x² + 3x + 1 = (x − 2)² → verified 7x − 3 = 0, not quadratic. (vii) (x + 2)³ = 2x(x² − 1) → verified the cubes do not cancel; what remains is −x³ + 6x² + 14x + 8 = 0, degree three, not quadratic. (viii) x³ − 4x² − x + 1 = (x − 2)³ → verified the cubes cancel and 2x² − 13x + 9 = 0 remains, quadratic. Items (vii) and (viii) are the sharpest pair in the exercise: both start as cubes on both sides, and only one of them survives the cancellation.
The history paragraph (§4.1, pp. 38–39), as a set of dated pegs: the Babylonians are credited with finding two positive numbers from their sum and product, a task the chapter says amounts to solving x² − px + q = 0; Euclid worked the problem as one about lengths; Brahmagupta (C.E. 598–665) set down a rule covering the type ax² + bx = c; Sridharacharya (C.E. 1025) is credited with the formula this chapter later uses, reached by completing the square and quoted by Bhaskara II; Al-Khwarizmi worked on the several types around C.E. 800; and Abraham bar Hiyya Ha-Nasi published full solutions in Europe in C.E. 1145 in Liber embadorum.

Figures to have open

Fig. 4.1 (p. 38), the labelled rectangle. Redraw as a clean schematic; the whole point is that one side is written in terms of the other, so the two labels and the area must be visible at once. Standard schematic — do not reproduce the printed artwork.
A standard-form annotation card for ax² + bx + c = 0 with call-outs on a, b, c and on the condition attached to a. Standard schematic.
A two-column sorting board for Exercise 4.1 Question 1, each item shown with its simplified equation underneath. Standard schematic.
A dated timeline strip for section 10. Standard schematic; the chapter prints no portraits or illustrations in the history paragraph.

Where this sits in the book

NCERT Mathematics, Textbook for Class X, Chapter 4 "Quadratic Equations", §4.1 Introduction, pp. 38–39, including the prayer-hall situation with Fig. 4.1 and the historical paragraph
Same chapter, §4.2 Quadratic Equations, pp. 39–41: the definition, the standard form, Example 1, Example 2 and the Remark
Same chapter, Exercise 4.1, pp. 41–42 (Question 1 is the sorting task used here; Question 2 belongs to the companion topic g10-maths-ch04-m01-t02)
Same chapter, §4.5 Summary, p. 47, point 1, which restates the definition
Backward pointer: Class X Chapter 2, for the quadratic polynomial and for degree