The slope of the line passing through points A and B is
slope = (b ² - a ²) / (b - a) = ((b - a) (b + a)) / (b - a) = b + a
Since this line passes through (0, 3), its equation would be
y - 3 = (b + a) (x - 0) ==> y = (b + a) x + 3
It also passes through (-3/2, 0), so that
y - 0 = (b + a) (x + 3/2) ==> y = (b + a) x + 3/2 (b + a)
If these equations describe the same line, then they must have the same slope and y-intercept, so that
3 = 3/2 (b + a) ==> a + b = 2
It also passes through (a, a ²), so that
a ² = (b + a) a + 3 ==> a ² = ab + a ² + 3 ==> ab = -3
Solving for b in the first underlined equation, we get
b = 2 - a
Substituting into the second equation and solving for a gives
a (2 - a) = -3 ==> 2a - a ² = -3 ==> a ² - 2a - 3 = (a - 3) (a + 1) = 0
==> a = 3 or a = -1
==> b = -1 or b = 3
Since a < b, we pick a = -1 and b = 3, so the solution is (a, b) = (-1, 3).