Final answer:
The equation initially provided can be rewritten as a quadratic in terms of 2^x. Using the quadratic formula's discriminant, we determine that the equation has two distinct real roots. Therefore, the correct answer is C, which corresponds to 2 real roots.
Step-by-step explanation:
The question asks for the number of real roots of the equation 5 + 2x - 1 = 2x(2x - 2). To find the real roots, we need to solve the equation for x. Let's simplify the equation, first by adding 1 to both sides to isolate the terms with exponents:
6 + 2x = 2x(2x - 2)
We then see that 2x appears on both sides of the equation, so we can try to cancel it out. However, if we subtract 2x from both sides, we get 6 = -2 * 2x, which implies 2x = -3. Since 2x cannot be negative, we understand that we have made an error in cancellation because 2x could be zero, making the equation undefined at that point.
Let's go back and rewrite the equation:
5 = 2x(2x - 2)
This is now a quadratic equation in terms of 2x. Let's set y = 2x and rewrite the equation as:
5 = y(y - 2)
Which can be rewritten as a standard quadratic equation:
y2 - 2y - 5 = 0
Using the quadratic formula, we find the value of y, and thus 2x. The original equation, as written, has only one term that could potentially be zero, which is 2x. So, solving for real y values, we will then solve 2x = y to find the real x values. However, as we are just determining the count of roots here and not their actual values, we can apply the discriminant of the quadratic equation y2 - 2y - 5 = 0. The discriminant, b2 - 4ac, will tell us the number of real roots: (2)2 - 4(1)(-5) = 4 + 20 = 24, which is greater than zero, indicating two distinct real roots.
Therefore, the correct answer is C, which corresponds to 2 real roots.