Question

The solutions to the equation f(x) = x^2 - 5x + 6 when f(2) and f(3) are the points ______

Answer 1

To find the solutions to the equation \( f(x) = x^2 - 5x + 6 \) when \( f(2) \) and \( f(3) \) are the points, we first need to calculate \( f(2) \) and \( f(3) \).

1. Calculate \( f(2) \):
\[
f(2) = 2^2 - 5 \cdot 2 + 6 = 4 - 10 + 6 = 0
\]

2. Calculate \( f(3) \):
\[
f(3) = 3^2 - 5 \cdot 3 + 6 = 9 - 15 + 6 = 0
\]

Both \( f(2) \) and \( f(3) \) equal 0, which indicates that both points are solutions of the equation \( f(x) = 0 \).

Now, we solve the equation \( x^2 - 5x + 6 = 0 \) to confirm the solutions.

This quadratic can be factored as:
\[
(x - 2)(x - 3) = 0
\]

Setting each factor to zero gives us:
1. \( x - 2 = 0 \) → \( x = 2 \)
2. \( x - 3 = 0 \) → \( x = 3 \)

Therefore, the solutions to the equation \( f(x) = x^2 - 5x + 6 = 0 \) are \( x = 2 \) and \( x = 3 \).

In conclusion, when \( f(2) \) and \( f(3) \) are evaluated, they yield the points \( (2, 0) \) and \( (3, 0) \), respectively. Thus, the points where \( f(x) = 0 \) are:
\[
(2, 0) \text{ and } (3, 0).
\]