To solve the problems, we first need to analyze the function \( f(x) = x^2 - 5x + 6 \).
Completing the Statements
1. The solutions to the equation f(x) = 6: To find when \( f(x) = 6 \), we set the equation equal to 6:
\[ x^2 - 5x + 6 = 6 \]
Subtracting 6 from both sides gives us:
\[ x^2 - 5x = 0 \]
Factoring out \( x \) results in:
\[ x(x - 5) = 0 \]
Thus, the solutions are:
\[ x = 0 \quad \text{or} \quad x = 5 \]
Therefore, the points where \( f(x) = 6 \) are \( (0, 6) \) and \( (5, 6) \).
2. The solutions to the equation f(x) = f(2) and f(3): Next, we find \( f(2) \) and \( f(3) \):
Calculating \( f(2) \):
\[ f(2) = 2^2 - 5 \cdot 2 + 6 = 4 - 10 + 6 = 0 \]
Calculating \( f(3) \):
\[ f(3) = 3^2 - 5 \cdot 3 + 6 = 9 - 15 + 6 = 0 \]
Both \( f(2) \) and \( f(3) \) equal 0. This means finding the points where \( f(x) = 0 \).
Returning to \( f(x) = 0 \):
\[ x^2 - 5x + 6 = 0 \]
Factoring gives us:
\[ (x - 2)(x - 3) = 0 \]
Thus, the solutions are:
\[ x = 2 \quad \text{or} \quad x = 3 \]
So, the points where \( f(x) = f(2) \) and \( f(3) \) are \( (2, 0) \) and \( (3, 0) \).
Summary of Answers
- The solutions to the equation \( f(x) = 6 \) are the points (0, 6) and (5, 6).
- The solutions to the equation \( f(x) = f(2) \) and \( f(3) \) are the points (2, 0) and (3, 0).
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