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Question:

What are the unique values of b and c in the equation 4x² + bx + c = 0 if one of the roots of the equation is (−1/2)?

A. The second root is 1/2
B. The ratio of c and b is 1

Explanation:

4x² + bx + c = 0                                                          …(i)

One root = −1/2

Using statement A alone:

Second root is 1/2.

Sum of the roots = −b/4 = (−1/2) + 1/2 = 0

∴ b = 0

Also, product of the roots = c/4 = (−1/2) × (1/2) = (−1/4)

∴ c = −1

So, statement A alone is sufficient to answer the question.

Using statement B alone:

Ratio of c and b is 1.

∴ b = c

∴ Equation (i) becomes 4x² + bx + b = 0                    ...(ii)

But one root = −1/2

Substituting x = −1/2 in the above equation (ii), we get,

1 − b/2 + b = 0

∴ b = −2 and c = −2

So, statement B alone is also sufficient to answer the question.

Hence, option (b).