Quadratic Equations
Quadratic Equations
Allen Star Batch
Grade 11
Question:
If $a,b,c \in \mathbb{R}, a > 10$ and $(x-a)(x-12) + 2 = (x+b)(x+c)$ for all $x \in \mathbb{R}$ then $|b-c| = $ _______.
Step-by-Step Solution
Key Concept: An integer constraint on roots reduces to finding integer factorizations of a constant term.
For $(x-a)(x-12) + 2 = 0$ to have integral roots, we need $(x-a)(x-12) = -2$. This means $(x-a)$ and $(x-12)$ are integer factor pairs of $-2$: either $-1 \times 2$, $1 \times (-2)$, $-2 \times 1$, or $2 \times (-1)$. Solving each case yields $a = 9$ or $a = 15$.
Correct Answer: 1