Let a and b be the solutions of the equation 2x^2+6x-14=0. What is the value of (2a-3)(4b-6)?
By the quadratic formula, the solutions are (-3 \pm sqrt(37))/2. Plugging them in, we get
(2(-3 + sqrt(37)/2 - 3)(4*(-3 - sqrt(37))/2 - 6) = -14
+2668 Note that\((2a - 3)(4b - 6) = 8ab - 12b - 12a + 18 = 8ab - 12(a + b) + 18\)
By Vieta, \(a + b = {-6 \over 2} = -3\) and \(ab = {-14 \over 2} = -7\)
Plugging these in gives us \({8 \times -7 - 12(-3) + 18} = \color{brown}\boxed{-2}\)
