Quadratic Equations
Quadratic Equations
Allen Star Batch
Grade 11
Question:
Let $P(x) = x^3 + ax^2 + bx^2 + cx + 1$ and $Q(x) = x^3 + cx^2 + bx + ax + 1$ with $a, b, c \in \mathbb{R}$ and $a \neq c$. If $P(x) = 0$ and $Q(x) = 0$ have two common roots then:
$b = -2$
$b = 2$
$a + c = 0$
$a - 2c = 0$
Step-by-Step Solution
Key Concept: Common roots of two polynomials satisfy both equations simultaneously, providing direct constraints on their coefficients.
If $P(x)$ and $Q(x)$ share a common factor $(x-a)(x-c)$, then $P(0) = Q(0)$ at the common roots. Solving shows $x = -1$ and $x = 1$ are the common roots, with $P(x) - Q(x) = (x-a-c)(x^2-1)$ uniquely determined.
Correct Answer: 1,3