The roots of 2x² + bx + c=0 are alpha and beta. If alpha=2beta. Show that 2b²=9c

Step 1: Identify the coefficients and apply Vieta's formulas.
For a quadratic equation $Ax^2 + Bx + C = 0$, the sum of the roots ($\alpha + \beta$) is $-\frac{B}{A}$ and the product of the roots ($\alpha \beta$) is $\frac{C}{A}$.
Given the equation $2x^2 + bx + c = 0$, we have $A=2$, $B=b$, and $C=c$.
So, the sum of the roots is:

$$\alpha + \beta = -\frac{b}{2} \quad (1)$$

And the product of the roots is:

$$$$