Class 10 Maths Chapter 4: Quadratic Equations

Quadratic Equations is Chapter 4 of NCERT Class 10 Maths. It teaches you to recognise a quadratic equation, find its roots by factorisation or by the quadratic formula, and predict the nature of the roots using the discriminant. Watch the six video lessons below, then go through these notes, formulas and solved examples to revise.

What you will learn

• The standard form of a quadratic equation and how to check whether an equation is quadratic
• Finding roots by splitting the middle term (factorisation)
• Using the quadratic formula to find roots of any quadratic equation
• The discriminant and what it tells you about the nature of roots
• Framing quadratic equations from word problems on numbers, areas and speed

Video Lessons

Quadratic Equations - Part 1

Quadratic Equations - Part 2

Quadratic Equations - Part 3

Quadratic Equations - Part 4

Quadratic Equations - Part 5

Quadratic Equations - Part 6

Key Concepts and Formulas

Standard Form

A quadratic equation is a polynomial equation of degree 2 in one variable. Before doing anything else, expand brackets and bring every term to one side so that the right side is zero.

ax² + bx + c = 0, where a ≠ 0

• If the x² terms cancel after simplifying, the equation is not quadratic (it becomes linear)
• A root is a value of x that makes ax² + bx + c equal to zero
• A quadratic equation has at most two roots

Roots by Factorisation

Factorisation is the quickest method when the numbers are friendly. The idea is to write the middle term as the sum of two terms whose product equals the product of the first and last terms.

Find two numbers with sum = b, product = a × c
(px + q)(rx + s) = 0
x = −q/p or x = −s/r

• Split bx using these two numbers, group in pairs, and take out common factors

Quadratic Formula

When the expression does not factorise easily, use the quadratic formula. It works for every quadratic equation with real roots, so it is a safe choice in the exam when you are stuck.

x = [−b ± √(b² − 4ac)] / 2a

• Always write a, b and c with their signs before substituting
• Simplify surds fully, for example √12 = 2√3

Discriminant and Nature of Roots

The expression under the square root in the formula decides what kind of roots the equation has. You can find it without solving the equation.

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

Word Problems

Questions on consecutive numbers, rectangles, ages, speed and time often lead to a quadratic equation. After solving, reject any root that does not make sense, such as a negative length or a negative speed.

Consecutive integers:          x, x + 1
Consecutive odd/even integers: x, x + 2
Area of rectangle = length × breadth
Time = distance ÷ speed

Solved Examples

Example 1: Is (x + 2)² = x² + 3x − 1 a quadratic equation?

x² + 4x + 4 = x² + 3x − 1
x² − x² + 4x − 3x + 4 + 1 = 0
x + 5 = 0 (degree 1)
∴ Not a quadratic equation (it is linear)

Example 2: Solve x² − 7x + 12 = 0 by factorisation.

Sum = −7, product = 12 → −3 and −4
x² − 3x − 4x + 12 = 0
x(x − 3) − 4(x − 3) = 0
(x − 3)(x − 4) = 0
∴ x = 3 or x = 4

Example 3: Solve 6x² − x − 2 = 0 by splitting the middle term.

a × c = 6 × (−2) = −12
−4 × 3 = −12 and −4 + 3 = −1
6x² − 4x + 3x − 2 = 0
2x(3x − 2) + 1(3x − 2) = 0
(2x + 1)(3x − 2) = 0
∴ x = −1/2 or x = 2/3

Example 4: Solve x² − 4x + 1 = 0 using the quadratic formula.

a = 1, b = −4, c = 1
D = (−4)² − 4 × 1 × 1 = 16 − 4 = 12
√12 = 2√3
x = (4 ± 2√3) / 2
∴ x = 2 + √3 or x = 2 − √3

Example 5: Find the nature of the roots of 3x² − 2x + 5 = 0.

a = 3, b = −2, c = 5
D = (−2)² − 4 × 3 × 5
D = 4 − 60 = −56
D < 0
∴ No real roots

Example 6: Find k if kx² − 12x + 9 = 0 has two equal roots.

For equal roots, D = 0
(−12)² − 4 × k × 9 = 0
144 − 36k = 0
∴ k = 4, giving (2x − 3)² = 0

Example 7: The length of a rectangular plot is 3 m more than its breadth and its area is 70 m². Find its dimensions.

Let breadth = b m, length = (b + 3) m
b(b + 3) = 70
b² + 3b − 70 = 0
(b + 10)(b − 7) = 0
b = 7 (b = −10 rejected, length cannot be negative)
∴ Breadth = 7 m, Length = 10 m

Common Mistakes to Avoid

• Applying the formula before writing the equation in standard form; a wrong sign on c changes the discriminant completely.
• Dividing both sides by x to "simplify", which loses the root x = 0. Factor out x instead.
• Writing −b² instead of b² in the discriminant; (−4)² is +16, not −16.
• Keeping a negative answer for a length, age or speed in a word problem; always state why you reject it to earn the full marks.

Quick Revision

Standard form: ax² + bx + c = 0, a ≠ 0
x = [−b ± √(b² − 4ac)] / 2a
D = b² − 4ac
D > 0 distinct real, D = 0 equal, D < 0 none
Equal roots: x = −b/2a

• Factorisation: split b into two numbers whose product is ac

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