The sum of all integral values of $k$ ($k \neq 0$) for which the equation $\frac{2}{x - 1} - \frac{1}{x - 2} = \frac{2}{k}$ in $x$ has no real roots, is _________.
Show Hint
When solving inequalities for integral values, always estimate the roots of the quadratic part to the nearest tenth to find the boundary integers accurately.
Step 1: Understanding the Concept:
We first simplify the algebraic equation into a quadratic form. For a quadratic equation $Ax^2 + Bx + C = 0$ to have no real roots, its discriminant $D = B^2 - 4AC$ must be less than zero. Step 2: Detailed Explanation:
Simplifying the given equation:
\[ \frac{2(x - 2) - (x - 1)}{(x - 1)(x - 2)} = \frac{2}{k} \]
\[ \frac{x - 3}{x^2 - 3x + 2} = \frac{2}{k} \]
\[ kx - 3k = 2x^2 - 6x + 4 \]
\[ 2x^2 - (6 + k)x + (4 + 3k) = 0 \]
For no real roots, $D<0$:
\[ (6 + k)^2 - 4(2)(4 + 3k)<0 \]
\[ 36 + 12k + k^2 - 32 - 24k<0 \]
\[ k^2 - 12k + 4<0 \]
The roots of $k^2 - 12k + 4 = 0$ are $k = \frac{12 \pm \sqrt{144 - 16}}{2} = \frac{12 \pm \sqrt{128}}{2} = 6 \pm 4\sqrt{2}$.
Since $4\sqrt{2} \approx 4 \times 1.414 = 5.656$, the range is:
\[ 6 - 5.656<k<6 + 5.656 \implies 0.344<k<11.656 \]
The integral values of $k$ in this range are $k \in \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}$.
Sum of these values $= \frac{11 \times 12}{2} = 66$. Step 3: Final Answer:
The sum of all such integral values of $k$ is 66.
