Question

The sum of all possible values of \(x\) satisfying the equation \(2^{4x^2} - 2^{2x^2+x+16} + 2^{2x+30} = 0\), is

• A. \(\frac{5}{2}\)
• B. \(\frac{1}{2}\)
• C. 3
• D. \(\frac{3}{2}\)

Answer 1

The Correct Option is B

Given Equation:
\(2^{4x^2} - 2^{2x^2+x+16} + 2^{2x+30} = 0\) 

First, rewrite the equation by breaking down the powers of 2 in terms of the base and their exponents:

\(\Rightarrow (2^{2x^2})^2 - 2^{2x^2} \cdot 2^{x+15} \cdot 2^1 + (2^{x+15})^2 = 0\)

This simplifies to:

\(\Rightarrow (2^{2x^2} - 2^{x+15})^2 = 0\)

Thus, we have the equation:

\(\Rightarrow 2^{2x^2} - 2^{x+15} = 0\)

Now, since the equation involves powers of 2, we can equate the exponents:

\(\Rightarrow 2^{2x^2} = 2^{x+15}\)

Equating the exponents of 2 gives us:

\(\Rightarrow 2x^2 = x + 15\)

Rearranging this into a standard quadratic form:

\(\Rightarrow 2x^2 - x - 15 = 0\)

Now, solve this quadratic equation by factoring:

\(\Rightarrow (2x + 5)(x - 3) = 0\)

Thus, the solutions for \(x\) are:

\(\Rightarrow x = -\frac{5}{2} \text{ or } x = 3\)

The sum of the possible values is:

\(-\frac{5}{2} + 3 = \frac{1}{2}\)

Therefore, the correct answer is (B): \(\frac{1}{2}\)