Asked by B
How many x-intercepts would the function shown below have?
f, of, x, equals, 2, left parenthesis, x, minus, 6, right parenthesis, left parenthesis, x, minus, 6, right parenthesis
f(x)=2(x−6)(x−6)
f, of, x, equals, 2, left parenthesis, x, minus, 6, right parenthesis, left parenthesis, x, minus, 6, right parenthesis
f(x)=2(x−6)(x−6)
Answers
Answered by
GPT-4o mini
To find the x-intercepts of the function \( f(x) = 2(x - 6)(x - 6) \), we set the function equal to zero:
\[
f(x) = 0
\]
This means we need to solve:
\[
2(x - 6)(x - 6) = 0
\]
Since \( 2 \) is a constant and does not affect the x-intercepts, we can simplify this to:
\[
(x - 6)(x - 6) = 0
\]
This simplifies to:
\[
(x - 6)^2 = 0
\]
The solution to this equation is:
\[
x - 6 = 0 \quad \Rightarrow \quad x = 6
\]
Since this factor is squared, the x-intercept at \( x = 6 \) has a multiplicity of 2. Thus, there is **one unique x-intercept** at \( x = 6 \).
In conclusion, the function \( f(x) = 2(x - 6)(x - 6) \) has **one x-intercept**.
\[
f(x) = 0
\]
This means we need to solve:
\[
2(x - 6)(x - 6) = 0
\]
Since \( 2 \) is a constant and does not affect the x-intercepts, we can simplify this to:
\[
(x - 6)(x - 6) = 0
\]
This simplifies to:
\[
(x - 6)^2 = 0
\]
The solution to this equation is:
\[
x - 6 = 0 \quad \Rightarrow \quad x = 6
\]
Since this factor is squared, the x-intercept at \( x = 6 \) has a multiplicity of 2. Thus, there is **one unique x-intercept** at \( x = 6 \).
In conclusion, the function \( f(x) = 2(x - 6)(x - 6) \) has **one x-intercept**.