Quadratic Equations

What the exams ask here

• find a^2+b^2, a^3+b^3 from roots
• one root is p times other
• k for equal roots
• HM of roots given
• 2x - x^2 = 0 integral solutions
• harmonic mean of the roots of (5 + sqrt2)x^2 - bx + 8 + 2sqrt5 = 0 is 4 (Indore 2023 Q19)
• roots of x^2 - px + 54 = 0 in the ratio 2:3 (JIPMAT 2024 Q19)
• 8x^2 - 2kx + k = 0 with one root p times the other (Indore 2025 MCQ Q24)
• 1/(a alpha + b) + 1/(a beta + b) (JIPMAT 2024 Q31)
• 7(y + 1/y) - 2(y^2 + 1/y^2) = 9: integral solutions (JIPMAT 2023 Q15, stored in number-system.integral-solutions)
• statements: (x^2 + 3x + 1) = (x - 2)^2 is not quadratic; x^2 + 2sqrt3 x + 3 = 0 has equal roots (JIPMAT 2023 Q16)
• x^2 - 12x + 35 = 0 and x^2 + ax + 105 = 0 share a root -> min of y^2 + 4y - 5a (Indore 2026 SA, memory-based)
• one student miswrites the constant, another the x-coefficient (JIPMAT 2025)
• one root between two numbers -> parameter range
• a root 2 + sqrt3 with rational coefficients