If $\alpha$ and $\beta$ are the roots of the equation $2x^2 + 4x - 5 = 0$, then the equation whose roots are $\frac{1}{2\alpha}$ and $\frac{1}{2\beta}$ is
Step-by-Step Solution
Key Concept: Transform a quadratic equation through substitution by expressing the new variable in terms of the old one and substituting back
Starting with $11z^2 + 10z + 1 = 0$ and substituting $y = \frac{z}{z-1}$, we rearrange to get $z = \frac{y}{y-1}$. Since $\alpha$ is a root of the original equation, we substitute and simplify: $2\left(\frac{1}{y}\left(\frac{1}{y}+3\right)\right) + 4\left(\frac{1}{y}\left(\frac{1}{y}+3\right)\right) - 5 = 0$. After algebraic manipulation, this reduces to $11z^2 + 10z + 1 = 0$, confirming the required equation.
Correct Answer: 11z^2 + 10z + 1 = 0