Need alpha? Don't get Uranium, find beta!

Substituting 4 in the first equation, we get $x^2 + \alpha x + 12 = 0$ $4^2 + 4 \alpha + 12 = 0$ $16 + 4 \alpha + 12 = 0$ $28 + 4 \alpha = 0$ $4 \alpha = -28$ $\boxed{\alpha = -7}$ Substituting the value of $\alpha$ in the second equation, we get $x^2 + \alpha x + \beta = 0$ $x^2 -7x + \beta = 0$ Since the roots are equal, we have $b^2 - 4ac = 0$ $(-7)^2 = 4(1)( \beta)$ $4 \beta = 49$ $\boxed{ \beta = \frac{49}{4}}$