Final answer:
The quadratic equation 2c² = 16c - 32 has one real solution, determined by rearranging it to standard form and applying the quadratic formula, which results in a zero discriminant.
Step-by-step explanation:
The equation given, 2c² = 16c - 32, is a quadratic equation of the form at² + bt + c = 0. To determine the type and number of solutions, we can first rearrange the equation to standard form by moving all terms to one side: 2c² - 16c + 32 = 0. Next, we can apply the quadratic formula to find the solutions. The quadratic formula is given by (-b ± √(b² - 4ac)) / (2a), where a, b, and c are constants from the quadratic equation at² + bt + c = 0. In our case, a = 2, b = -16, and c = 32. We calculate the discriminant, Δ = b² - 4ac, to determine the nature of the solutions. Here, Δ = (-16)² - 4(2)(32) = 256 - 256 = 0. Since the discriminant is zero, the equation has one real solution.