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MathsMediumClass 11

Common Roots

Quadratic Equations

7

JEE Qs

8%

Hard

50

min

Master the systematic derivation of common root conditions and always analyze both cases (one vs. two common roots) based on problem wording.

🧮 Key Formulas

Let two quadratic equations be a_1x^2 + b_1x + c_1 = 0 and a_2x^2 + b_2x + c_2 = 0.
If they have ONE common root α, then a_1α^2 + b_1α + c_1 = 0 and a_2α^2 + b_2α + c_2 = 0.
Using cross-multiplication on these two equations for α^2 and α:
α^2 / (b_1c_2 - b_2c_1) = α / (c_1a_2 - c_2a_1) = 1 / (a_1b_2 - a_2b_1)
From this, α = (c_1a_2 - c_2a_1) / (a_1b_2 - a_2b_1)
The condition for ONE common root is (b_1c_2 - b_2c_1)(a_1b_2 - a_2b_1) = (c_1a_2 - c_2a_1)^2
If they have TWO common roots, then the two quadratic equations are essentially the same (or multiples of each other). The condition is: a_1/a_2 = b_1/b_2 = c_1/c_2

✅ Key Points for JEE

  • 1Always distinguish between problems asking for ONE common root and TWO common roots, as the conditions are different.
  • 2For one common root, solve the system of two equations by substituting the common root (say, 'α') into both, then eliminate 'α^2' or 'α' to find the common root or establish the condition.
  • 3The cross-multiplication method provides a systematic way to derive the common root (α) and the condition for its existence.
  • 4When using the cross-multiplication method, carefully check the denominator (a_1b_2 - a_2b_1). If it's zero, it implies a special case where the coefficients are proportional, potentially leading to two common roots or no common roots, requiring separate analysis.
  • 5If the two equations have two common roots, they must be scalar multiples of each other, meaning their corresponding coefficients are proportional.

⚠️ Common Mistakes

  • ✕Assuming the common root condition automatically means only one common root, without considering the possibility of two common roots.
  • ✕Making algebraic errors while solving the system of equations for α or applying the cross-multiplication formula.
  • ✕Forgetting to check for cases where `a_1b_2 - a_2b_1 = 0` when using the formulas derived from cross-multiplication, as it can lead to division by zero.
  • ✕Not verifying if the given equations are indeed quadratic for all parameter values (i.e., if the leading coefficient can be zero).

📝 Practice Questions

See all

Q22.The roots of the quadratic equation 3x2 −px + q = 0 are 10th and 11th terms of an arithmetic progression with common difference 32 . If the sum of the first 11 terms of this arithmetic progression is 88 , then q −2p is equal to -.

2025·NumericalMedium

Q13. The number of real solution(s) of the equation x2 + 3x + 2 = min{|x −3|, |x + 2|} is : (1) 1 (2) 0 (3) 2 (4) 3

2025·MCQMedium

Q17.Let αθ and βθ be the distinct roots of 2x2 + (cos θ)x −1 = 0, θ ∈(0, 2π). If m and M are the minimum and the maximum values of α4θ + β4θ , then 16(M + m) equals : (1) 24 (2) 25 (3) 17 (4) 27

2025·MCQHard

Q22.If the equation a(b −c)x2 + b(c −a)x + c(a −b) = 0 has equal roots, where a + c = 15 and b = 365 , then a2 + c2 is equal to

2025·NumericalMedium

Q7. Let the line passing through the points (−1, 2, 1) and parallel to the line x−12 = y+13 = 4z intersect the line y−3 x+2 3 = 2 = z−41 at the point P . Then the distance of P from the point Q(4, −5, 1) is (1) 5 (2) 5√5 (3) 5√6 (4) 10

2025·MCQMedium

Q13.The sum, of the squares of all the roots of the equation x2 + |2x −3| −4 = 0, is (1) 3(3 −√2) (2) 6(3 −√2) (3) 6(2 −√2) (4) 3(2 −√2)

2025·MCQMedium

NCERT Chapters

  • Class 11 Maths Ch 5: Complex Numbers and Quadratic Equations