What are the unique values of b and c in the equation 4x^2 + bx + c = 0 if one of the roots of the equation is (-1/2) ?
I. The second root is 1/2
II. The ratio of c and b is 1
Here is my take.
In the question statement, we are given 4x^2 + bx + c = 0 and one of the roots of the equation is (-1/2). This means that the given quadratic formula has to be simplified into one of the following:
(4x + 2)(x + c/2) where b = 2 + 2c
(2x + 1)(2x + c) where b = 2 + 2c
(x + 1/2)(4x + 2c) where b = 2 + 2c
therefore we need either b or c to get the unique value of b and c.
if the other root is equal to 1/2 then:
(1/2 + c/2) = 0 so c = -1
(2(1/2) + c) = 0 so c = -1
(4(1/2) + 2c) = 0 so c = -1
and b = 0 so sufficient
b/c = 1 so b = c.
since we know from the question that b = 2 + 2c,
b = 2 + 2b
so b = -2 which is also equal to c
So D is my final answer.