Final answer:
To find the minimum value of the quadratic function f(x) = x² - 6x + 2, we complete the square to rewrite it as (x - 3)² - 7, revealing that the minimum value is -7 at x = 3.
Step-by-step explanation:
The student has asked about finding the minimum value of a quadratic function, which is a problem in Mathematics, specifically within algebra. The function given is f(x) = x² - 6x + 2. To find the minimum value of this function, we can complete the square to convert it into vertex form.
We start with the quadratic term and the linear term: x² - 6x. To complete the square, we take half of the coefficient of x, which is -6, divide by 2 to get -3, and then square it to get 9. We add and subtract this number inside the function: f(x) = (x² - 6x + 9) - 9 + 2. This can be rewritten as: f(x) = (x - 3)² - 7.
Now, in the form f(x) = (x - h)² + k, we can see that the vertex of the parabola is at (h, k), which in this case is (3, -7). Since the coefficient of the x² term is positive, the parabola opens upwards, and the vertex represents the minimum value. Therefore, the minimum value of f(x) is -7 at x = 3.