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

The function given is \( f(x) = x^2 - 5x + 6 \).

1. **Finding the solutions to the equation when \( f(x) = 6 \)**:
We want to solve \( f(x) = 6 \):

\[
x^2 - 5x + 6 = 6
\]

Simplifying this:

\[
x^2 - 5x + 6 - 6 = 0 \implies x^2 - 5x = 0
\]

Factoring:

\[
x(x - 5) = 0
\]

Therefore, the solutions are:

\[
x = 0 \quad \text{or} \quad x = 5
\]

Thus, **the solutions to the equation \( f(x) = 6 \) are the points** \((0, 6)\) and \((5, 6)\).

2. **Finding the solutions to the equation when \( f(2) \) and \( f(3) \)**:
First, we evaluate \( f(2) \) and \( f(3) \):

\[
f(2) = 2^2 - 5(2) + 6 = 4 - 10 + 6 = 0
\]
\[
f(3) = 3^2 - 5(3) + 6 = 9 - 15 + 6 = 0
\]

Therefore:

- The solution to the equation when \( f(2) = 0 \) is the point \((2, 0)\).
- The solution to the equation when \( f(3) = 0 \) is the point \((3, 0)\).

Thus, **the solutions to the equation when \( f(2) \) and \( f(3) \) are the points** \((2, 0)\) and \((3, 0)\).