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SAT - Math - Quadratic Equations

If (ax+2)(bx+7) = 15x2 + cx +14 for all values of x, and a + b = 8, what are the two possible values for c?

If (ax 2)(bx 7) = 15x2 cx 14 for all values of x, and a b = 8, what are the two possible values for c?

(ax+2)(bx+7) = 15x2 + cx +14 

expanding LHS

abx2 + (2b + 7a)x + 14 = 15x2 + cx + 14

ab = 15 ... (1)

2b + 7a = c ... (2)

given: a + b = 8 ... (3)

from (1) and (3)

b = 15/a

a + 15/a = 8

a2 - 8a + 15 = 0

a2 - 5a - 3a + 15 = 0

a(a - 5) - 3(a - 5) = 0

(a - 3) (a - 5) =0

a = 3, a = 5

and

b = 5, b = 3

Case 1:

c = 2b + 7a = 2*3 + 7*5 = 41

Case 2

c = 2b + 7a = 2*5 + 7*3 = 31