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The sum of all the real values of $$x$$ satisfying the equation $$2^{(x-1)(x^2+5x-50)} = 1$$ is:
We begin with the given exponential equation
$$2^{(x-1)(x^2+5x-50)} = 1.$$
The base $$2$$ is positive and unequal to $$1$$. For a positive base other than $$1$$, an expression of the form $$a^k$$ equals $$1$$ only when its exponent is zero. Therefore we must have
$$(x-1)(x^2+5x-50)=0.$$
This product is zero exactly when at least one of its factors is zero. We analyze each factor in turn.
First factor.
$$x-1=0 \;\;\Longrightarrow\;\; x=1.$$
Second factor.
$$x^2+5x-50=0.$$
To solve the quadratic, compute its discriminant:
$$\Delta = b^2 - 4ac = 5^2 - 4(1)(-50) = 25 + 200 = 225.$$
The square root of the discriminant is
$$\sqrt{\Delta} = \sqrt{225} = 15.$$
Using the quadratic formula,
$$x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-5 \pm 15}{2(1)}.$$
This yields two real roots:
$$x = \frac{-5 + 15}{2} = \frac{10}{2} = 5,$$
$$x = \frac{-5 - 15}{2} = \frac{-20}{2} = -10.$$
Collecting all real solutions, we have
$$x = -10,\; 1,\; 5.$$
The question asks for the sum of all these real values:
$$\text{Sum} = (-10) + 1 + 5.$$
Perform the additions step by step:
$$-10 + 5 = -5,$$
$$-5 + 1 = -4.$$
Thus the required sum equals $$-4$$.
Hence, the correct answer is Option C.
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