Quadratic Equations
Quadratic Equations
Allen Star Batch
Grade 11
Question:
The solution of the equation $\frac{8}{[x]} - \frac{9}{x} + \frac{10}{\{x\}} = 2$ is of the form $\frac{k+1}{k}, k \in \mathbb{N}$ then $k = $ _______. ($[x]$ denotes largest integer less than or equal to $x$, and $\{x\}$ denotes fractional part of $x$)
Step-by-Step Solution
Key Concept: Decomposing $x$ into integer and fractional parts and using constraints on the fractional part restricts the integer part to specific values.
Let $[x] = t$ and $\{x\} = y$ where $0 \leq y < 1$ and $x = t + y$ with $x \neq 0$. The equation becomes $8t^2 = 11ty + 10y^2 < 11t + 10$, which simplifies to $8t^2 - 11t - 10 < 0$. Factoring gives $-\frac{5}{8} < t < 2$. Since $t$ is an integer and $x \neq 0$, we have $t = 1$, yielding $y = \frac{1}{2}$.
Correct Answer: 2