Answer:
±12 (two answers)
Explanation:
Suppose one root is a. Then the other root will be -3a. The product of the two roots is the ratio of the constant coefficient to the leading coefficient:
(a)(-3a) = -27/4
a² = -27/(4·(-3)) = 9/4
a = ±√(9/4) = ±3/2
Then the other root is
-3a = -3(±3/2) = ±9/2 . . . . . . the roots will have opposite signs
We know the opposite of the sum of these roots will be the ratio of the linear term coefficient to the leading coefficient: b/4, so ...
-(a + (-3a)) = b/4
2a = b/4
b = 8a = 8·(±3/2)
b = ±12
_____
Check
For b = 12, the equation factors as ...
4x² +12x -27 = (2x -3)(2x +9) = 0
It has roots -9/2 and +3/2, the ratio of which is -3.
For b = -12, the equation factors as ...
4x² -12x -27 = (2x +3)(2x -9) = 0
It has roots 9/2 and -3/2, the ratio of which is -3.
