Identify roots

Let the original roots be $\alpha$ and $\beta$. We are given $\alpha + \beta = \frac{11}{2}$ and $\alpha \beta = \frac{15}{2}$.

$\alpha + \beta = \frac{11}{2}, \alpha \beta = \frac{15}{2}$

Determine new roots

The new equation has roots $2\alpha$ and $2\beta$. The new sum of roots is $2\alpha + 2\beta = 2(\alpha + \beta) = 2 \times \frac{11}{2} = 11$.

$Sum_{new} = 11$

Calculate new product

The new product of roots is $(2\alpha)(2\beta) = 4(\alpha \beta) = 4 \times \frac{15}{2} = 30$.

$Product_{new} = 30$

Construct the equation

Using the standard form $x^2 - (Sum_{new})x + Product_{new} = 0$, substitute the values calculated.

$x^2 - 11x + 30 = 0$