If the sum of the roots of the quadratic equation $ky^2 - 11y + (k - 23) = 0$ is $\frac{13}{21}$ more than the product of the roots, then find the value of $k$.
Generated by claude-sonnet-4-6 · 2026-06-15 10:33 · grounding rag
Model Answer
For $ky^2 - 11y + (k-23) = 0$:
- Sum of roots $= \dfrac{11}{k}$
- Product of roots $= \dfrac{k-23}{k}$
Given: Sum = Product $+ \dfrac{13}{21}$
$$\frac{11}{k} = \frac{k-23}{k} + \frac{13}{21}$$
$$\frac{11}{k} - \frac{k-23}{k} = \frac{13}{21}$$
$$\frac{11 - k + 23}{k} = \frac{13}{21}$$
$$\frac{34 - k}{k} = \frac{13}{21}$$
$$21(34 - k) = 13k$$
$$714 - 21k = 13k$$
$$714 = 34k$$
$$k = 21$$
Explanation
- Use Vieta's formulas: for $ax^2+bx+c=0$, sum of roots $= -b/a$ and product $= c/a$.
- Set up the equation directly from the given condition and solve for $k$.
- This is a standard application question; show each algebraic step clearly to earn both marks.