If a and b are the roots of the quadratic equation 2x2 + ( α − 3 ) x + ( α 2 − 2 ) then, maximum possible value of a2 + b2 is
-
5
-
–5
-
3 4
-
None
Roots a, b satisfy a+b = -(a-3)/2 and ab = (a^2-2)/2. a^2+b^2 = (a+b)^2 - 2ab = ((a-3)/2)^2 - (a^2-2) = (a^2-6a+9)/4 - a^2 + 2 = (-3a^2-6a+17)/4. This is a downward parabola with maximum at a = -b/2a = 6/(-6) = -1. Max value = (-3(1) - 6(-1) + 17)/4 = 20/4 = 5.
Using the sum and product of roots formulas for the equation two x squared plus the quantity alpha minus three times x plus the quantity alpha squared minus two equals zero, we find the sum of the roots a plus b equals the quantity three minus alpha divided by two, and the product a times b equals the quantity alpha squared minus two divided by two. The sum of their squares, a squared plus b squared, equals the quantity a plus b squared minus two times a times b, which simplifies to negative one fourth times alpha squared plus three halves times alpha plus one fourth. This expression represents a downward opening parabola, so its maximum value occurs at the vertex where alpha equals negative b divided by two a, giving alpha equals three. Substituting alpha equals three into the expression gives negative one fourth times nine plus three halves times three plus one fourth, which equals five. Thus, the maximum possible value of a squared plus b squared is five.