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Quadratic Equations ā€” Nature of Roots + Graph

Quadratic Equations

36

JEE Qs

12%

Hard

75

min

Always analyze the sign of 'a' and the value of the discriminant 'D' thoroughly, as they dictate both the nature of roots and the orientation and position of the quadratic graph, crucial for problem-solving.

šŸ§® Key Formulas

D = b^2 - 4ac
For ax^2 + bx + c = 0, roots x = (-b +/- sqrt(D)) / 2a
Vertex of parabola y = ax^2 + bx + c is (-b/(2a), -D/(4a))
Axis of symmetry of parabola is x = -b/(2a)

āœ… Key Points for JEE

  • 1The discriminant D determines the nature of roots: D > 0 for real & distinct roots, D = 0 for real & equal roots, D < 0 for imaginary & conjugate roots.
  • 2If coefficients a, b, c are rational, then for D > 0, roots are rational if D is a perfect square, and irrational if D is not a perfect square.
  • 3The graph of y = ax^2 + bx + c is a parabola opening upwards if a > 0 and downwards if a < 0. Its x-intercepts are the roots of the equation.
  • 4The vertex (-b/(2a), -D/(4a)) represents the minimum value of the quadratic if a > 0 (range: [-D/(4a), infinity)) and the maximum value if a < 0 (range: (-infinity, -D/(4a)]).
  • 5For ax^2 + bx + c > 0 for all x: a > 0 and D < 0. For ax^2 + bx + c < 0 for all x: a < 0 and D < 0.

āš ļø Common Mistakes

  • āœ•Confusing conditions for 'real roots' (D >= 0) with 'rational roots' (D must be a perfect square).
  • āœ•Incorrectly determining the direction of the parabola (up/down) or the location of its vertex, leading to errors in graph sketching or range determination.
  • āœ•Forgetting to consider the coefficient 'a' in conditions for sign of quadratic expression for all x (e.g., ax^2+bx+c > 0 for all x requires a > 0 and D < 0).
  • āœ•Assuming 'a' cannot be zero in problems where it's a variable or parameter, leading to missing cases (e.g., if a term (k-1)x^2 is given, k=1 might make it a linear equation).

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 11 Maths Ch 5: Quadratic Equations