Question
f(x)=x^2-5x+6 on a coordinate plane
The solutions to the equation f(x)=x^2-5x+6 when f(x)=6 are the points ______
The solutions to the equation f(x)=x^2-4x*6 when f(2) and f(3) are the points ____
The solutions to the equation f(x)=x^2-5x+6 when f(x)=6 are the points ______
The solutions to the equation f(x)=x^2-4x*6 when f(2) and f(3) are the points ____
Answers
GPT-4o mini
Let's solve each part of your question step by step.
### Part 1: Finding the solutions to \( f(x) = x^2 - 5x + 6 \) when \( f(x) = 6 \).
First, let's express \( f(x) \):
\[
f(x) = x^2 - 5x + 6
\]
Now, we want to find when \( f(x) = 6 \):
\[
x^2 - 5x + 6 = 6
\]
Subtracting 6 from both sides gives:
\[
x^2 - 5x + 6 - 6 = 0 \Rightarrow x^2 - 5x = 0
\]
Factoring out an \( x \):
\[
x(x - 5) = 0
\]
Setting each factor to zero gives:
\[
x = 0 \quad \text{or} \quad x - 5 = 0 \Rightarrow x = 5
\]
Thus, the solutions when \( f(x) = 6 \) are the points \( (0, 6) \) and \( (5, 6) \).
### Part 2: Finding the solutions to the equation \( f(x) = x^2 - 4x \cdot 6 \) when \( f(2) \) and \( f(3) \).
First, we need to evaluate \( f(2) \) and \( f(3) \) using the function \( f(x) = x^2 - 5x + 6 \).
Calculating \( f(2) \):
\[
f(2) = 2^2 - 5 \cdot 2 + 6 = 4 - 10 + 6 = 0
\]
Calculating \( f(3) \):
\[
f(3) = 3^2 - 5 \cdot 3 + 6 = 9 - 15 + 6 = 0
\]
Thus, we find that:
\[
f(2) = 0 \quad \text{and} \quad f(3) = 0
\]
It seems there may be a typo in the equation \( f(x) = x^2 - 4x \cdot 6 \). Assuming you meant \( f(x) = 0 \), we already know that both \( 2 \) and \( 3 \) yield \( f(x) = 0 \).
Thus, the points for \( f(2) \) and \( f(3) \) corresponding to \( f(x) = 0 \) are \( (2, 0) \) and \( (3, 0) \).
### Final Answers:
- The solutions to the equation \( f(x) = 6 \) are the points **(0, 6) and (5, 6)**.
- The solutions to the equation at \( f(2) \) and \( f(3) \) are the points **(2, 0) and (3, 0)**.
### Part 1: Finding the solutions to \( f(x) = x^2 - 5x + 6 \) when \( f(x) = 6 \).
First, let's express \( f(x) \):
\[
f(x) = x^2 - 5x + 6
\]
Now, we want to find when \( f(x) = 6 \):
\[
x^2 - 5x + 6 = 6
\]
Subtracting 6 from both sides gives:
\[
x^2 - 5x + 6 - 6 = 0 \Rightarrow x^2 - 5x = 0
\]
Factoring out an \( x \):
\[
x(x - 5) = 0
\]
Setting each factor to zero gives:
\[
x = 0 \quad \text{or} \quad x - 5 = 0 \Rightarrow x = 5
\]
Thus, the solutions when \( f(x) = 6 \) are the points \( (0, 6) \) and \( (5, 6) \).
### Part 2: Finding the solutions to the equation \( f(x) = x^2 - 4x \cdot 6 \) when \( f(2) \) and \( f(3) \).
First, we need to evaluate \( f(2) \) and \( f(3) \) using the function \( f(x) = x^2 - 5x + 6 \).
Calculating \( f(2) \):
\[
f(2) = 2^2 - 5 \cdot 2 + 6 = 4 - 10 + 6 = 0
\]
Calculating \( f(3) \):
\[
f(3) = 3^2 - 5 \cdot 3 + 6 = 9 - 15 + 6 = 0
\]
Thus, we find that:
\[
f(2) = 0 \quad \text{and} \quad f(3) = 0
\]
It seems there may be a typo in the equation \( f(x) = x^2 - 4x \cdot 6 \). Assuming you meant \( f(x) = 0 \), we already know that both \( 2 \) and \( 3 \) yield \( f(x) = 0 \).
Thus, the points for \( f(2) \) and \( f(3) \) corresponding to \( f(x) = 0 \) are \( (2, 0) \) and \( (3, 0) \).
### Final Answers:
- The solutions to the equation \( f(x) = 6 \) are the points **(0, 6) and (5, 6)**.
- The solutions to the equation at \( f(2) \) and \( f(3) \) are the points **(2, 0) and (3, 0)**.