The roots of ax^2 + bx + c = 0 are real and positive. alpha and beta are real. Then ax^2 + b|x| + c = 0 has: A. no roots B. 2 real roots C. 3 real roots D. 4 real roots

GATE 2014 · General Aptitude · Quadratic Equations · medium

Answer: ax^2 + b|x| + c = 0 has 4 real roots: -q, -p, +p, +q (option D)

1. Analyse sign conditions from positive roots: Both roots positive => sum > 0 => -b/a > 0; product > 0 => c/a > 0. Taking a > 0: b < 0 and c > 0.
2. Substitute y = |x| into the equation: For y = |x| >= 0, the equation becomes ay^2 + by + c = 0, which is identical to the original quadratic. Its roots are p and q (both positive), so y = p and y = q are the two valid solutions.
3. Recover all x from |x| = p and |x| = q: |x| = p => x = p or x = -p (2 roots); |x| = q => x = q or x = -q (2 roots). Total = 4 distinct real roots.