Quadratic Equations – Class 10 Maths Chapter 4 Notes & NCERT PDF

Class 10MathematicsChapter 4NCERT book: Mathematics - Class 10

Notes and a simple summary of Chapter 4 of the NCERT Class 10 Mathematics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

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Chapter 4: Quadratic Equations

Introduction

The chapter opens with a real problem: a charitable trust plans a prayer hall of carpet area 300 m² whose length is one metre more than twice its breadth. If the breadth is x m, then x(2x + 1) = 300, i.e. 2x² + x − 300 = 0 — a quadratic equation. The book notes that Indian mathematicians such as Brahmagupta gave a method for solving equations of the form ax² + bx = c, and Sridharacharya derived the formula now known as the quadratic formula.

Key Concepts & Theorems

A quadratic equation in x has the form ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0. This is its standard form.

A real number α is a root of ax² + bx + c = 0 if aα² + bα + c = 0. The zeroes of the polynomial ax² + bx + c and the roots of the equation are the same.

A quadratic equation has at most two roots.

Checking whether an equation is quadratic

Simplify completely first. For example, (x − 2)² + 1 = 2x − 3 becomes x² − 6x + 8 = 0 (quadratic), but x(x + 1) + 8 = (x + 2)(x − 2) becomes x + 12 = 0 (not quadratic, as the x² terms cancel).

Quadratic formula and discriminant

Quadratic formula: the roots of ax² + bx + c = 0 are x = (−b ± √(b² − 4ac)) / 2a, provided b² − 4ac ≥ 0.

D = b² − 4ac is the discriminant. The equation has:

two distinct real roots if b² − 4ac > 0,

two equal real roots if b² − 4ac = 0 (each root is −b/2a),

no real roots if b² − 4ac < 0.

Methods Step by Step

Solving by factorisation (splitting the middle term)

Write the equation in standard form.

Find two numbers whose product is a × c and sum is b.

Split bx into two terms, group and factorise into two linear factors.

Put each factor equal to zero; reject roots that do not fit the situation (e.g. negative lengths).

Finding the nature of roots

Identify a, b, c.

Compute D = b² − 4ac.

Decide using the sign of D; if D ≥ 0 and roots are asked for, use the formula.

Worked Examples

1. 2x² − 5x + 3 = 0. Product ac = 6, sum −5: split as −2x − 3x → (2x − 3)(x − 1) = 0, roots 1 and 3/2.

2. 6x² − x − 2 = 0. (3x − 2)(2x + 1) = 0, roots 2/3 and −1/2.

3. 3x² − 2√6x + 2 = 0. This is (√3x − √2)² = 0, so the root √(2/3) is repeated (two equal roots).

4. The prayer hall. 2x² + x − 300 = 0 → 2x² − 24x + 25x − 300 = 0 → (x − 12)(2x + 25) = 0. Breadth cannot be negative, so x = 12 m and length 2x + 1 = 25 m.

5. Discriminant. For 2x² − 4x + 3 = 0: D = 16 − 24 = −8 < 0, so there are no real roots. For 3x² − 2x + 1/3 = 0: D = 4 − 4 = 0, two equal roots 1/3.

6. The pole problem. A circular park has diameter 13 m; a pole on the boundary must be 7 m farther from gate A than from gate B (diametrically opposite). With BP = x, AP = x + 7 and (angle in a semicircle) x² + (x + 7)² = 13², i.e. x² + 7x − 60 = 0. D = 49 + 240 = 289 > 0; x = 5 (reject −12). So the pole is 5 m from B and 12 m from A.

Other book problems: two numbers with sum 27 and product 182 are 13 and 14; consecutive positive integers whose squares add to 365 are 13 and 14; a right triangle with hypotenuse 13 cm and altitude 7 cm less than the base has base 12 cm and altitude 5 cm; a mango grove with length twice the breadth and area 800 m² is possible — 40 m × 20 m; a park with perimeter 80 m and area 400 m² is possible — a 20 m × 20 m square.

Board Exam Focus

1 mark: identify a quadratic equation; state the nature of roots from D.

2 marks: find k so that roots are equal (set b² − 4ac = 0).

3–5 marks: frame a quadratic from a word problem (areas, ages, speed, numbers), solve, and reject the impossible root with a reason.

Show the splitting of the middle term clearly — marks are given for factorisation and for the final roots.

Formula Sheet

Standard form: ax² + bx + c = 0, a ≠ 0

x = (−b ± √(b² − 4ac)) / 2a

D = b² − 4ac; D > 0 two distinct real roots, D = 0 two equal real roots (−b/2a), D < 0 no real roots

Sum of roots −b/a, product c/a (from Chapter 2)

Common Mistakes

Not simplifying before deciding whether an equation is quadratic.

Sign errors in b² − 4ac, especially when b is negative (b² is always non-negative).

Keeping a negative length, age or number of objects as an answer.

Forgetting that "two equal roots" still means the equation has real roots.

💡 Key Learning Points

✓Recognise a quadratic equation ax² + bx + c = 0 (a ≠ 0) after simplifying, and know its roots are the zeroes of ax² + bx + c.
✓Solve quadratic equations by splitting the middle term (factorisation).
✓Use the quadratic formula x = (−b ± √(b² − 4ac))/2a.
✓Use the discriminant b² − 4ac to decide whether roots are distinct real, equal real or not real.
✓Model real situations (prayer hall, pole in a park, numbers, triangles) as quadratics and reject roots that do not make sense.

👨‍🏫 Teaching Tips

→Open with the prayer-hall problem and let students set up 2x² + x − 300 = 0 themselves.
→Practise 'product ac, sum b' number hunts as a warm-up before factorising.
→Show that x(x + 1) + 8 = (x + 2)(x − 2) is not quadratic, to stress simplifying first.
→Connect the discriminant to graphs: a parabola cutting, touching or missing the x-axis.
→Use the pole-in-the-park problem to link geometry (angle in a semicircle) with algebra.
→Have students write a one-line reason whenever they reject a root.
→Mention Brahmagupta and Sridharacharya to show the Indian roots of the quadratic formula.

📋 Assessment Questions

Can the student solve 6x² − x − 2 = 0 by splitting the middle term?
Can the student find the discriminant of 2x² − 4x + 3 = 0 and state the nature of its roots?
Can the student frame and solve the pole problem, rejecting the negative root?
Can the student find k for which kx² + ... has equal roots by setting D = 0?
Can the student decide whether a park of perimeter 80 m and area 400 m² is possible?

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