Let a,b,c be positive integers in arithmetic progression such that the equation $\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}=0$ has only integer solutions. Then which of the following statements is (are) TRUE ? (A) $\mathrm{c}-\mathrm{b}$ is an integer multiple of a (B) Both the roots of the equation $\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}=0$ are odd integers (C) If $\mathrm{c}=15$, then $\mathrm{ab}=8$ (D) If $\mathrm{b}=8$, then $\mathrm{x}=3$ is a root of the equation $\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}=0$

(A) $\mathrm{c}-\mathrm{b}$ is an integer multiple of a; (B) Both the roots of the equation $\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}=0$ are odd integers; (C) If $\mathrm{c}=15$, then $\mathrm{ab}=8$

(D) If $\mathrm{b}=8$, then $\mathrm{x}=3$ is a root of the equation $\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}=0$

JEE Advanced · 2026 · 17 May 2026 · Paper 2 · Quadratic Equations and Inequalities

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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 ?
(A)

c - b

is an integer multiple of a
(B) Both the roots of the equation

ax^{2} + bx + c = 0

are odd integers
(C) If

c = 15

, then

ab = 8

(D) If

b = 8

, then

x = 3

is a root of the equation

ax^{2} + bx + c = 0

(A) $c - b$ is an integer multiple of a✓ Correct
(B)Both the roots of the equation $ax^{2} + bx + c = 0$ are odd integers✓ Correct
(C)If $c = 15$ , then $ab = 8$ ✓ Correct
(D)If $b = 8$ , then $x = 3$ is a root of the equation $ax^{2} + bx + c = 0$

Answer: (A), (B), (C)

Solution

Since

a, b, c

are in AP:

2 b = a + c

. Let the integer roots be

α, β

. Then

a x^{2} + b x + c = a (x - α) (x - β)

. Putting

x = - 2

4 a - 2 b + c = a (α + 2) (β + 2)

. Since

2 b = a + c

, we get

4 a - a - c + c = a (α + 2) (β + 2) \Rightarrow 3 a = a (α + 2) (β + 2) \Rightarrow (α + 2) (β + 2) = 3

. Since

a, b, c > 0

, both roots are negative. Thus

(α + 2) (β + 2) = (- 1) \times (- 3)

(- 3) \times (- 1)

, giving

(α, β) = (- 3, - 5)

(- 5, - 3)

. From

2 b = a + c

and Vieta's formulas:

b = 8 a

and

c = 15 a

. Options A, B, C can be verified. For option D: if

b = 8

, then

a = 1, c = 15

, and the equation is

x^{2} + 8 x + 15 = 0

with roots

- 3, - 5

, not 3.

JEE Advanced · 2026 · 17 May 2026 · Paper 2 · One or more correct options

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JEE Advanced · 2026 · 17 May 2026 · Paper 2 · Quadratic Equations and Inequalities

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