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Quadratic equations show up across Class 10 and keep coming back in Class 11 and in JEE. The good news is that the Class 10 chapter is compact. This post covers what is in it now, how to factorise reliably, when to switch to the formula, how to read the discriminant, and the mistakes that cost marks.
What does the Class 10 chapter include now?
The current NCERT chapter (2026-27 reprint) has four main sections: introduction, quadratic equations, solution by factorisation, and nature of roots, where the quadratic formula is stated. There is no separate section on solving by completing the square. CBSE's Class 10 Mathematics curriculum lists the standard form ax² + bx + c = 0 (a ≠ 0), solutions "by factorization, and by using quadratic formula", the link between the discriminant and nature of roots, and situational problems. Check your own year's curriculum before the exam.
How do you factorise by splitting the middle term?
For ax² + bx + c = 0, find two numbers whose product is ac and whose sum is b.
Worked example 1. Solve 2x² − 7x + 3 = 0.
ac = 2 × 3 = 6. You need two numbers with product 6 and sum −7: −6 and −1.
Split: 2x² − 6x − x + 3 = 0.
Group: 2x(x − 3) − 1(x − 3) = 0, so (2x − 1)(x − 3) = 0.
Roots: x = 1/2 or x = 3.
Check: 2(3)² − 7(3) + 3 = 18 − 21 + 3 = 0. Always substitute one root back.
When should you use the quadratic formula?
Use it when you cannot spot the factor pair within a minute, or when the roots are not whole numbers. The formula is x = [−b ± √(b² − 4ac)] / 2a, valid when b² − 4ac ≥ 0.
Worked example 2 (word problem). The product of two consecutive positive integers is 306. Find them.
Let the integers be n and n + 1. Then n(n + 1) = 306, so n² + n − 306 = 0.
a = 1, b = 1, c = −306. b² − 4ac = 1 + 1224 = 1225, and √1225 = 35.
n = (−1 ± 35)/2, so n = 17 or n = −18.
The question says positive, so reject −18. The integers are 17 and 18. Check: 17 × 18 = 306.
Same example by formula, cross-checked: 2x² − 7x + 3 = 0 gives b² − 4ac = 49 − 24 = 25, so x = (7 ± 5)/4 = 3 or 1/2. Both methods agree.
How does the discriminant tell you the nature of roots?
The NCERT chapter calls b² − 4ac the discriminant. The equation has:
two distinct real roots if b² − 4ac > 0,
two equal real roots if b² − 4ac = 0,
no real roots if b² − 4ac < 0.
Worked example 3. For what value of k does kx² + 6x + 1 = 0 have two equal roots?
Equal roots need b² − 4ac = 0: 36 − 4k = 0.
So k = 9.
Check: 9x² + 6x + 1 = (3x + 1)², which has the single repeated root x = −1/3.
NCERT's own example, 2x² − 4x + 3 = 0, has discriminant 16 − 24 = −8, so it has no real roots.
What mistakes cost marks?
Not rearranging to zero. x² + 3x = 10 must become x² + 3x − 10 = 0 before you factorise.
Dividing by x. From x² = 5x, dividing by x gives only x = 5 and loses x = 0. Write x² − 5x = 0, so x(x − 5) = 0, and x = 0 or 5.
Sign of b in the formula. If b = −7, then −b = +7. Write a, b and c with their signs before substituting.
Forgetting to reject a root. In word problems, lengths, ages and counts cannot be negative. Say why you reject the root.
Writing 2a only under the square root. The whole numerator, −b ± √(b² − 4ac), is divided by 2a.
A ten-day practice routine
Days 1 to 3: ten factorisation questions a day, with a substitution check for one root of each.
Days 4 and 5: the same equations by formula; confirm both methods agree.
Days 6 and 7: discriminant questions only, including "find k" types.
Days 8 and 9: word problems on numbers, areas, speed and age; write the equation in words first.
Day 10: a timed mixed set of ten, then sort your errors by the list above.
The chapter's introduction mentions that the Indian mathematician Sridharacharya derived the formula, and that Al-Khwarizmi studied different types of quadratic equations. Read how Al-Khwarizmi solved them in words in our post on the origins of algebra. Because roots are the zeroes of a quadratic polynomial, our polynomials post for Class 10 is a good companion. When a word problem will not turn into an equation, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning.
Start now: solve 3x² − 5x + 2 = 0 by factorisation and by the formula, and check that you get the same two roots.
