Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Advanced2026MathematicsQuadratic EquationActual

Let a, b, c be positive integers in arithmetic progression such that the equation ax^2 + bx + c = 0 has only integer solutions. Then which of the following statements is (are) TRUE ?

Options

  1. Ac - b is an integer multiple of a
  2. BBoth the roots of the equation ax^2 + bx + c = 0 are odd integers
  3. CIf c = 15 , then ab = 8
  4. DIf b = 8 , then x = 3 is a root of the equation ax^2 + bx + c = 0

Correct answer

A. c - b is an integer multiple of a

Step-by-step solution

Given a, b, c are in arithmetic progression, we have 2b = a + c . Let the integer roots of the equation ax^2 + bx + c = 0 be and . Sum of the roots is + = - b a and product of the roots is = c a . Dividing the arithmetic progression condition by a , we get: 2b a = 1 + c a Substituting the sum and product of the roots: -2( + ) = 1 + + 2 + 2 + 1 = 0 Adding 3 to both sides to factorize: + 2 + 2 + 4 = 3 ( + 2)( + 2) = 3 Since and are integers, ( + 2) and ( + 2) must be integer factors of 3 . The possible pairs are (1,

Practice Quadratic Equation on Quantrex Academy →

More from Quadratic Equation

Match each entry in List-I to the correct entry in List-II and choose the correct option. List-I List-II (P) If and are the distinct roots of the equation x^2 + x + 1 = 0 , then th 2026Consider the quadratic equation (n^2 - 2n + 2)x^2 - 3x + (n^2 - 2n + 2)^2 = 0 , n R . Let be the minimum value of the product of its roots and be the maximum value of the sum of it 2026Let one root of the quadratic equation in x : (k^2 - 15k + 27)x^2 + 9(k-1)x + 18 = 0 be twice the other. Then the length of the latus rectum of the parabola y^2 = 6kx is equal to: 2026Let e₁ and e₂ be two distinct roots of the equation x^2 - ax + 2 = 0 . Let the sets a R : e₁ and e₂ are the eccentricities of hyperbolas = ( , ) , and a R : e₁ and e₂ are the eccen 2026Let , be the roots of the equation x^2 - x + p = 0 and , be the roots the equation x^2 - 4x + q = 0 ; p, q Z . If , , , are in G.P., then |p + q| equals : 2026Let a, b C . Let , be the roots of the equation x^2 + ax + b = 0 . If - = 11 and ^2 - ^2 = 3i 11 , then ( ^3 - ^3)^2 is equal to: 2026Let A, B , where A, B (- 2 , 2 ) , be the roots of the quadratic equation x^2 - 2x - 5 = 0 . Then 20 ^2 ( A+B 2 ) is equal to: 2026If =1 and =1+i 2 , where i= -1 are two roots of the equation x^3+ax^2+bx+c=0 , a,b,c R , then _ -1 ¹(x^3+ax^2+bx+c)dx is equal to: 2026 Full Quadratic Equation list All JEE Advanced PYQs