Let a and b be the solutions of the equation 2x2−10x+5=0. What is the value of (2a−3)(4b−6)?
The solutions to 2x^2 - 10x + 5 = 0 are (5 +/- sqrt(15))/2 Plugging these values in, you get
(2*(5 + sqrt(15))/2 - 3)(4*(5 - sqrt(15))/2 - 6) = -18
(2a - 3) ( 4b - 6) =
(2a - 3) (2) (2b - 3) =
2 ( 2a - 3) (2b - 3) =
2 [ 4ab - 6a - 6b + 9 ]
2 [ 4ab - 6 ( a+ b) + 9 ]
The product of the roots = 5/2
The sum of the roots = - (-10) / 2 = 5
So
2 [ 4 (5/2) - 6 ( 5) + 9 ] =
2 [ 10 - 30 + 9 ] =
2 [ -11 ] =
-22
Another solution! :D
P(x) = 2x^2 - 10x + 5 = 2(x - a)(x - b)
(2a - 3)(4b - 6) = 8(a - 3/2)(b - 3/2)
P(3/2) = 2(3/2 - a)(3/2 - b) = -11/2
-11/2 * 4 = -22.
