Question

{"given":"We have the quadratic equation $x^2 - 2kx + k^2 + k - 5 = 0$ and need to find the values of $k$ for which both roots are less than 5. Let $f(x) = x^2 - 2kx + k^2 + k - 5$. ","key_observation":"For both roots to be less than 5, we need three conditions: (1) The discriminant must be non-negative for real roots, (2) $f(5) > 0$ since both roots are to the left of $x = 5$, and (3) The vertex (axis of symmetry) should be less than 5 to ensure both roots are on the same side.","option_analysis":[{"label":"(A)","text":"(5, 6]","verdict":"incorrect","explanation":"This interval suggests $k > 5$, but when $k > 5$, the condition $f(5) > 0$ leads to $(k-4)(k-5) > 0$, which gives $k > 5$ or $k < 4$. However, combined with discriminant condition $k ≤ 5$, this creates a contradiction."},{"label":"(B)","text":"(6, ∞)","verdict":"incorrect","explanation":"This interval is completely outside the valid range. When $k > 6$, the discriminant condition $k ≤ 5$ is violated, so real roots don't exist, making this option invalid."},{"label":"(C)","text":"(-∞, 4)","verdict":"correct","explanation":"From discriminant condition: $k ≤ 5$. From $f(5) > 0$: $k^2 - 9k + 20 > 0$, giving $(k-4)(k-5) > 0$, so $k < 4$ or $k > 5$. Combining with $k ≤ 5$, we get $k < 4$. The axis of symmetry is at $x = k$, and for both roots less than 5, we need $k < 5$, which is satisfied."},{"label":"(D)","text":"[4, 5]","verdict":"incorrect","explanation":"When $k = 4$, we have $f(5) = 0$, meaning one root equals 5, violating the condition that both roots are strictly less than 5. When $4 < k ≤ 5$, $f(5) < 0$, which means both roots are not less than 5."}],"answer":"(C)","formula_steps":[]}