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Why u said that x to the power 2 minus X plus 1 is greater or equal than 3/4

The inequality \( x^2 - x + 1 \geq \frac{3}{4} \) is used to show that the expression \( x^2 - x + 1 \) is always positive for all real values of \( x \), ensuring there are no division by zero issues.

Let’s break down why this inequality holds:

1. **Starting point**: Consider the quadratic expression \( x^2 - x + 1 \). We need to prove that it is always greater than or equal to \( \frac{3}{4} \).

2. **Complete the square**: 
   To simplify \( x^2 - x + 1 \), we can complete the square:
   \[
   x^2 - x + 1 = \left(x - \frac{1}{2}\right)^2 + \frac{3}{4}
   \]
   This shows that \( x^2 - x + 1 \) can be written as the sum of a perfect square \( \left(x - \frac{1}{2}\right)^2 \) and the constant \( \frac{3}{4} \).

3. **Non-negative square**: Since \( \left(x - \frac{1}{2}\right)^2 \) is always non-negative (a square of any real number is always greater than or equal to zero), it follows that:
   \[
   \left(x - \frac{1}{2}\right)^2 \geq 0
   \]
   
4. **Conclusion**: This implies that:
   \[
   \left(x - \frac{1}{2}\right)^2 + \frac{3}{4} \geq \frac{3}{4}
   \]
   Therefore, \( x^2 - x + 1 \geq \frac{3}{4} \) for all real values of \( x \).

This inequality ensures that the denominator \( x^2 - x + 1 \) never becomes zero, as it is always greater than or equal to \( \frac{3}{4} \), thus avoiding division by zero.

Would you like to explore any of the following related questions or concepts?

1. How to complete the square for quadratic expressions?
2. What is the geometric interpretation of completing the square?
3. Can a quadratic expression have a negative value?
4. What does the discriminant tell us about the behavior of a quadratic function?
5. How can we find the minimum value of a quadratic expression?

**Tip**: When dealing with quadratic expressions, completing the square is a powerful tool to analyze their behavior, especially when determining minimum or maximum values.

Why did you say that x² - x + 1 is greater than or equal to 3/4?

Explore why the quadratic expression x² - x + 1 is always greater than or equal to 3/4 through the method of completing the square. This solution explains how the expression remains positive for all real values of x, avoiding division by zero issues. Suitable for high school students.

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