CBSE • Class 10 • Mathematics • Chapter 4 (Quadratic Equations) • Exercise 4.1

Exercise 4.1: Quadratic Equations — NCERT Solutions

Standard form, identifying quadratic equations, and solving by factorisation.

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What this exercise covers

Standard formFactorisationRoots from factors

Step-by-step solutions — Exercise 4.1

2 solved questions • Each solution includes a Socratic hint, full working and a common-mistake callout • Last reviewed 2026-09-03

Exercise 4.1 Q1 • 2 marks

Check whether the equation (x−3)(2x+1)=x(x+5)(x-3)(2x+1) = x(x+5) is a quadratic equation. If so, express it in the standard form.
Hint (Socratic — try this first)▾
After expanding both sides, does the highest power of xx remain 22?
Step-by-step solution▾

Expand the left side: (x−3)(2x+1)=2x2+x−6x−3=2x2−5x−3(x-3)(2x+1) = 2x^2 + x - 6x - 3 = 2x^2 - 5x - 3.

Expand the right side: x(x+5)=x2+5xx(x+5) = x^2 + 5x.

Bring all terms to one side: 2x2−5x−3−x2−5x=02x^2 - 5x - 3 - x^2 - 5x = 0 x2−10x−3=0x^2 - 10x - 3 = 0

The highest power of xx is 22 and the coefficient of x2x^2 is 1≠01 \neq 0. Hence it is a quadratic equation in standard form x2−10x−3=0x^2 - 10x - 3 = 0.

Common mistake:
Students often forget to move ALL terms to one side, or make a sign error while transposing x2+5xx^2 + 5x, leaving the equation not truly simplified.
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Exercise 4.1 Q2 • 3 marks

Solve the quadratic equation 6x2−x−2=06x^2 - x - 2 = 0 by the method of factorisation.
Hint (Socratic — try this first)▾
Can you split the middle term into two numbers whose product is 6×(−2)6\times(-2) and whose sum is −1-1?
Step-by-step solution▾

We split the middle term. We need two numbers whose product is 6×(−2)=−126 \times (-2) = -12 and whose sum is −1-1. These numbers are −4-4 and 33.

6x2−4x+3x−2=06x^2 - 4x + 3x - 2 = 0 2x(3x−2)+1(3x−2)=02x(3x - 2) + 1(3x - 2) = 0 (3x−2)(2x+1)=0(3x - 2)(2x + 1) = 0

Setting each factor to zero: 3x−2=0⇒x=233x - 2 = 0 \Rightarrow x = \tfrac{2}{3} 2x+1=0⇒x=−122x + 1 = 0 \Rightarrow x = -\tfrac{1}{2}

Thus the roots are x=23x = \dfrac{2}{3} and x=−12x = -\dfrac{1}{2}.

Common mistake:
Choosing factors that give the wrong product sign (e.g. taking +12+12 instead of −12-12) so the split does not reproduce the original middle term.
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How to approach Exercise 4.1

  1. Re-read the chapter summary first. Open Quadratic Equations and refresh the key concepts: Standard form, Factorisation, Quadratic formula, Discriminant.
  2. Try each problem yourself before opening the solution. Ten focused minutes per problem usually beats reading two finished solutions.
  3. Use Guru AI for the questions you get stuck on. The Socratic AI tutor walks you through it with hints instead of dictating the answer.
  4. Mark the questions you got wrong and revisit them after 24 hours — the spacing is what locks the method into long-term memory.

All exercises in Quadratic Equations

  1. Exercise 4.1 — Standard form, identifying quadratic equations, and solving by factorisation.
  2. Exercise 4.2 — Solving quadratic equations using the quadratic formula and completing the square.
  3. Exercise 4.3 — Nature of roots — using the discriminant to classify roots without solving.

Exercise 4.2 →

Solving quadratic equations using the quadratic formula and completing the square.

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