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MathsMediumClass 11

Nature of Roots ā€” Discriminant, types

Quadratic Equations

7

JEE Qs

8%

Hard

60

min

Always scrutinize the given constraints on the coefficients (real, rational, integer) as they are pivotal for correctly classifying the nature of roots, particularly when `D > 0`.

šŸ§® Key Formulas

ax^2 + bx + c = 0 (where a != 0)
D = b^2 - 4ac
If a, b, c are real coefficients:
D > 0 => Roots are real and distinct
D = 0 => Roots are real and equal
D < 0 => Roots are non-real (imaginary) and distinct (conjugate pairs)
If a, b, c are rational coefficients (a != 0):
D > 0 and D is a perfect square => Roots are rational and distinct
D > 0 and D is NOT a perfect square => Roots are irrational and distinct (conjugate pairs)
D = 0 => Roots are rational and equal
D < 0 => Roots are non-real (imaginary) and distinct (conjugate pairs)

āœ… Key Points for JEE

  • 1The discriminant `D = b^2 - 4ac` is the sole determinant of the nature of roots for a quadratic equation `ax^2 + bx + c = 0` (where `a != 0`).
  • 2Always identify the nature of coefficients (real, rational, integer) of the quadratic equation before classifying roots as rational/irrational or complex.
  • 3For real coefficients, if `D < 0`, the roots are always non-real (complex) and occur in conjugate pairs (e.g., `p + iq, p - iq`).
  • 4For rational coefficients, if `D > 0` and `D` is not a perfect square, the roots are always irrational and occur in conjugate pairs (e.g., `p + sqrt(q), p - sqrt(q)`).
  • 5If any of the coefficients `a, b, c` are irrational or complex, the standard rules for conjugate pairs for irrational/imaginary roots may not apply directly; the quadratic formula must be used carefully.

āš ļø Common Mistakes

  • āœ•Failing to consider the nature of coefficients (rational vs. real) when determining if roots are rational/irrational.
  • āœ•Making calculation errors in computing the discriminant `D = b^2 - 4ac`, especially sign errors.
  • āœ•Confusing the condition `D > 0` (real and distinct roots) with the condition that `D` must be a perfect square (rational and distinct roots).
  • āœ•Incorrectly assuming irrational or complex roots always form conjugate pairs if coefficients are not strictly rational or real, respectively.

šŸ“ Practice Questions

See all

Q22.The roots of the quadratic equation 3x2 āˆ’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 āˆ’2p is equal to -.

2025Ā·NumericalMedium

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

2025Ā·MCQMedium

Q17.Let Ī±Īø and Ī²Īø be the distinct roots of 2x2 + (cos Īø)x āˆ’1 = 0, Īø āˆˆ(0, 2Ļ€). If m and M are the minimum and the maximum values of Ī±4Īø + Ī²4Īø , then 16(M + m) equals : (1) 24 (2) 25 (3) 17 (4) 27

2025Ā·MCQHard

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

2025Ā·NumericalMedium

Q7. Let the line passing through the points (āˆ’1, 2, 1) and parallel to the line xāˆ’12 = y+13 = 4z intersect the line yāˆ’3 x+2 3 = 2 = zāˆ’41 at the point P . Then the distance of P from the point Q(4, āˆ’5, 1) is (1) 5 (2) 5āˆš5 (3) 5āˆš6 (4) 10

2025Ā·MCQMedium

Q13.The sum, of the squares of all the roots of the equation x2 + |2x āˆ’3| āˆ’4 = 0, is (1) 3(3 āˆ’āˆš2) (2) 6(3 āˆ’āˆš2) (3) 6(2 āˆ’āˆš2) (4) 3(2 āˆ’āˆš2)

2025Ā·MCQMedium

NCERT Chapters

  • Class 10 Mathematics Ch 4: Quadratic Equations
  • Class 11 Mathematics Ch 5: Complex Numbers and Quadratic Equations

šŸš€ This Unlocks

Common Roots