Given Equation:
\(2^{4x^2} - 2^{2x^2+x+16} + 2^{2x+30} = 0\)
Rewrite the equation by decomposing the exponential terms:
\(\Rightarrow (2^{2x^2})^2 - 2^{2x^2} \cdot 2^{x+15} \cdot 2^1 + (2^{x+15})^2 = 0\)
This simplifies to:
\(\Rightarrow (2^{2x^2} - 2^{x+15})^2 = 0\)
Which means:
\(\Rightarrow 2^{2x^2} - 2^{x+15} = 0\)
Since the equation involves powers of the same base (2), we can equate the exponents:
\(\Rightarrow 2^{2x^2} = 2^{x+15}\)
Equating the exponents yields:
\(\Rightarrow 2x^2 = x + 15\)
Rearrange into standard quadratic form:
\(\Rightarrow 2x^2 - x - 15 = 0\)
Solve the quadratic equation by factoring:
\(\Rightarrow (2x + 5)(x - 3) = 0\)
The possible values for \(x\) are:
\(\Rightarrow x = -\frac{5}{2} \text{ or } x = 3\)
The sum of these possible values is:
\(-\frac{5}{2} + 3 = \frac{1}{2}\)
Thus, the correct answer is (B): \(\frac{1}{2}\)