To find the minimum value of the function \( f(x) = x^2 - 2x + 3 \), we can complete the square or use calculus. Let's go step by step with the completion of the square:

1. The given function is:
   \[
   f(x) = x^2 - 2x + 3
   \]

2. We want to complete the square. First, take the coefficient of \( x \) (which is -2), divide it by 2, and square it:
   \[
   \left(\frac{-2}{2}\right)^2 = 1
   \]

3. Add and subtract 1 inside the expression:
   \[
   f(x) = (x^2 - 2x + 1) + 3 - 1
   \]
   \[
   f(x) = (x - 1)^2 + 2
   \]

Now, the function is in the form \( (x - 1)^2 + 2 \). Since \( (x - 1)^2 \) is always non-negative (its minimum value is 0), the minimum value of \( f(x) \) occurs when \( (x - 1)^2 = 0 \), which happens when \( x = 1 \).

4. At \( x = 1 \), the minimum value of the function is:
   \[
   f(1) = 0 + 2 = 2
   \]

Thus, the minimum value of \( f(x) = x^2 - 2x + 3 \) is **2**.

Do you want further details, or have any questions?

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Here are 5 related questions:
1. What is the vertex of a parabola in standard form \( ax^2 + bx + c \)?
2. How can you find the minimum or maximum of a quadratic function using derivatives?
3. What is the axis of symmetry for the function \( f(x) = x^2 - 2x + 3 \)?
4. Can you determine the roots of the equation \( x^2 - 2x + 3 = 0 \)?
5. How does completing the square help in converting a quadratic equation to vertex form?

**Tip:** The vertex of a parabola given by \( ax^2 + bx + c \) is found at \( x = -\frac{b}{2a} \).

Learn how to find the minimum value of the quadratic function x^2 - 2x + 3 using the completing the square method. This step-by-step solution covers algebraic concepts and theorems relevant to students in Grades 9-11.

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