To determine for which real numbers x the expression x² – 6x + 9 is negative, we first need to analyze the quadratic equation. We can start by rewriting the expression:
x² – 6x + 9 can be factored as (x – 3)². This means that the expression represents a perfect square.
The key property of a perfect square is that it is always non-negative; that is, (x – 3)² >= 0 for all real numbers x. The expression equals 0 when x = 3, and it is positive for all other real numbers.
Therefore, the expression x² – 6x + 9 is:
- Zero when x = 3
- Positive for all x ≠ 3
Since there are no values of x for which the expression is negative, we conclude that there are no real numbers x such that x² – 6x + 9 is negative.