If $ \alpha, \beta $ are the roots of equation $(k + 1) x^{2} - (20k + 14)x + 91k + 40 = 0 ; where \ (\alpha < \beta)$, $k > 0$, then The smaller root $\alpha$ lies in the interval

Treating the equation as a quadratic in k, we rearrange it to x^2(k - 20k + 91k) + x^2 - 14x + 40 - k + k = 0, which simplifies to (x - 4)(x - 10) + k(x^2 - 20x + 91) = 0. This shows the curve f(x) = (x - 4)(x - 10) intersects the curve g(x) = -k(x^2 - 20x + 91), which has roots at 7 and 13. Since the roots of the original equation are separated by 4, 7, 10, and 13, and the smaller root alpha must be to the left of the smallest root of g(x) because k > 0, alpha lies in the interval (4, 7).

• Thanks
• Wrong explanation