Question

The graph of the equation f(x) = x2 − 5x + 6
is shown on the coordinate plane.



Complete the statements

(2 points)
The solutions to the equation f(x) = x2 − 5x + 6
when f(x) = 6 are the points

The solutions to the equation f(x) = x2 − 5x + 6
when f(2) and f(3) are the points

Answer 1

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
1. The solutions to the equation \( f(x) = 6 \) are the points **(0, 6) and (5, 6)**.
2. The solutions to the equation \( f(x) = f(2) \) and \( f(3) \) are the points **(2, 0) and (3, 0)**.