1. **State the problem:** We have two quadratic equations: - Equation 1: $px^2 - 12x + 10 = 0$ with sum of roots = 4. - Equation 2: $px^2 - 42x + pk = 0$ has two equal roots. We need to find the value of $k$. 2. **Use the sum of roots formula for Equation 1:** For a quadratic $ax^2 + bx + c = 0$, sum of roots $= -\frac{b}{a}$. Here, $a = p$, $b = -12$, so sum of roots $= -\frac{-12}{p} = \frac{12}{p}$. Given sum of roots $= 4$, so: $$\frac{12}{p} = 4$$ 3. **Solve for $p$:** $$12 = 4p$$ $$p = \frac{12}{4} = 3$$ 4. **Use the condition for equal roots in Equation 2:** For equal roots, discriminant $\Delta = b^2 - 4ac = 0$. Here, $a = p = 3$, $b = -42$, $c = pk = 3k$. Calculate discriminant: $$\Delta = (-42)^2 - 4 \times 3 \times 3k = 1764 - 36k$$ Set $\Delta = 0$: $$1764 - 36k = 0$$ 5. **Solve for $k$:** $$36k = 1764$$ $$k = \frac{1764}{36} = 49$$ **Final answer:** $$k = 49$$