By the fundamental theorem of algebra, we can write the given quadratic in terms of its roots a and b as
3x ² - 5x + k = 3 (x - a) (x - b)
and upon expanding, we have
3x ² - 5x + k = 3x ² - (a + b) x + ab
Two polynomials are equivalent if their coefficients are equal, so
a + b = 5
ab = k
We're given that |a - b| = 2. If we assume a > b (that is, a is the larger root) then |a - b| = a - b = 2, or b = a - 2. Substitute this for b in the first equation above and solve for a :
a + (a - 2) = 5
2a - 2 = 5
2a = 7
a = 7/2
Solve for b :
b = 7/2 - 2
b = 3/2
Solve for k :
k = 7/2 × 3/2 = 21/4