Quadratic Equations
Quadratic Equations
star_batch_jee_advanced_2025
Grade 11
Question:
Let $f(x) = x^2 + bx + c, b,c \in \mathbb{R}$. If $f(x)$ is a factor of both $x^4 + 6x^2 + 25$ and $3x^4 + 4x^2 + 28x + 5$, then the minimum value of $f(x)$ is _______.
Step-by-Step Solution
Key Concept: Finding a common factor by taking linear combinations of two polynomials simplifies the problem significantly.
We observe that $3(x^4 + 6x^2 + 25) - (3x^4 + 4x^2 + 28x + 5) = 14(x^2 - 2x + 5)$. Thus $f(x) = x^2 - 2x + 5 = (x-1)^2 + 4$ is a common factor, and $f(x)$ must divide both given polynomials.
Correct Answer: 4