NCERT Solutions: Quadratic Equations (Class 10 Maths)

Chapter overview

A quadratic equation is a polynomial equation of degree 2, written in standard form as ax² + bx + c = 0, where a ≠ 0. This chapter covers three methods to find its roots and how to determine the nature of those roots.

The three solving methods

1. Factorisation — split the middle term so the equation factors into two linear terms.
2. Completing the square — rewrite the equation as a perfect square plus a constant.
3. Quadratic formula — x = [-b ± √(b² - 4ac)] / 2a

Nature of roots (the discriminant)

The discriminant D = b² - 4ac tells you the nature of the roots without solving fully:

• D > 0: two distinct real roots
• D = 0: two equal real roots (repeated root)
• D < 0: no real roots (roots are complex/imaginary)

Worked practice problem 1 — Factorisation

Question: Solve x² - 7x + 12 = 0 by factorisation.

Solution: We need two numbers that multiply to 12 and add to -7: those are -3 and -4.

x² - 3x - 4x + 12 = 0
x(x - 3) - 4(x - 3) = 0
(x - 3)(x - 4) = 0

Answer: x = 3 or x = 4

Worked practice problem 2 — Quadratic formula

Question: Solve 2x² + 5x - 3 = 0 using the quadratic formula.

Solution: Here a = 2, b = 5, c = -3.

D = b² - 4ac = 25 - 4(2)(-3) = 25 + 24 = 49
x = [-5 ± √49] / 4 = [-5 ± 7] / 4

Answer: x = 1/2 or x = -3

Worked practice problem 3 — Word problem

Question: The product of two consecutive positive integers is 240. Find the integers.

Solution: Let the integers be x and x + 1.

x(x + 1) = 240
x² + x - 240 = 0

Using the quadratic formula with a = 1, b = 1, c = -240:

D = 1 + 960 = 961, √961 = 31
x = (-1 ± 31) / 2

x = 15 (rejecting the negative root since the integers must be positive).

Answer: The integers are 15 and 16.

Common mistakes to avoid

• Forgetting to check that a ≠ 0 before calling an equation quadratic.
• Sign errors when substituting negative values of b or c into the discriminant formula.
• In word problems, forgetting to reject a root that doesn’t make sense in context (e.g. a negative length or a negative “positive integer”).

Looking for the official textbook PDF for this chapter? Visit our Class 10 NCERT Books page for the direct download link.