Question

The solutions to the equation f(x) = x^2 − 5x + 6
when f(x) = 6 are the points
(2,0) , (3,0) , (4,2) , (1,2) , (0,6) , (5,6)

Answer 1

To solve the equation \( f(x) = x^2 - 5x + 6 \) when \( f(x) = 6 \), we set the equation equal to 6:

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

Subtracting 6 from both sides gives:

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

Simplifying this, we have:

\[
x^2 - 5x = 0
\]

Factoring out an \( x \):

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

Setting each factor equal to zero gives the solutions:

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

So, the solutions for \( x \) when \( f(x) = 6 \) are \( x = 0 \) and \( x = 5 \).

Next, we can find the corresponding \( y \)-values by substituting back into \( f(x) \):

- For \( x = 0 \):

\[
f(0) = 0^2 - 5(0) + 6 = 6
\]
So the point is \( (0, 6) \).

- For \( x = 5 \):

\[
f(5) = 5^2 - 5(5) + 6 = 6
\]
So the point is \( (5, 6) \).

Thus, the points you listed as solutions do include \( (0, 6) \) and \( (5, 6) \).

The points (2, 0), (3, 0), (4, 2), and (1, 2) do not correspond to \( f(x) = 6 \) and are not required for this solution.

The final correct points that you listed when \( f(x) = 6 \) are indeed \( (0, 6) \) and \( (5, 6) \).