Question

Which expression is equivalent to 1/2 x^2 + 2?

A) (1/2)(x+i)(x-i)

B) (1/2)(x+2)(x-2)

C) (1/2)(x+2i)(x-2i)

D) (1/2)(x+1)(x-1)

Answer 1

The equivalent expression to 1/2 x^2 + 2 is option C, which is (1/2)(x+2i)(x-2i), as it correctly expands to the given expression including the imaginary unit i.

Step-by-step explanation:

The expression 1/2 x^2 + 2 can be thought of as a modified version of a perfect square trinomial. Considering each option given, we need to determine which factored form will expand to the original expression.

1. Option A: (1/2)(x+i)(x-i): When expanded, results in 1/2(x^2 - i^2). However, since i represents the imaginary unit where i^2 = -1, the expression simplifies to 1/2 x^2 + 1/2 which is not equivalent to the original expression.
2. Option B: (1/2)(x+2)(x-2): When expanded, results in 1/2(x^2 - 4), which simplifies to 1/2 x^2 - 2, again not equivalent to the original expression.
3. Option C: (1/2)(x+2i)(x-2i): When expanded, this uses the difference of squares factoring to become 1/2(x^2 - (2i)^2), or 1/2 x^2 + 2, which matches the original expression.
4. Option D: (1/2)(x+1)(x-1): When expanded, results in 1/2(x^2 - 1), which is 1/2 x^2 - 1/2, not equivalent to the original expression.

Therefore, the equivalent expression to 1/2 x^2 + 2 is Option C: (1/2)(x+2i)(x-2i).