Certainly! Let's find the values of [tex]\( t \)[/tex] given the function [tex]\( f(x) = 5 + x^2 \)[/tex] and the condition [tex]\( f(3 - t) = 9 \)[/tex].
1. First, substitute [tex]\( 3 - t \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[
f(3 - t) = 5 + (3 - t)^2
\][/tex]
2. Expand and simplify the expression inside the function:
[tex]\[
(3 - t)^2 = 9 - 6t + t^2
\][/tex]
So:
[tex]\[
f(3 - t) = 5 + 9 - 6t + t^2 = 14 - 6t + t^2
\][/tex]
3. Set this expression equal to 9 based on the given condition [tex]\( f(3 - t) = 9 \)[/tex]:
[tex]\[
14 - 6t + t^2 = 9
\][/tex]
4. Rearrange the equation to form a standard quadratic equation:
[tex]\[
t^2 - 6t + 14 - 9 = 0
\][/tex]
Simplify:
[tex]\[
t^2 - 6t + 5 = 0
\][/tex]
5. Solve the quadratic equation [tex]\( t^2 - 6t + 5 = 0 \)[/tex]. Factorize it if possible:
[tex]\[
t^2 - 6t + 5 = (t - 1)(t - 5) = 0
\][/tex]
6. Solve for the values of [tex]\( t \)[/tex]:
[tex]\[
t - 1 = 0 \quad \Rightarrow \quad t = 1
\][/tex]
[tex]\[
t - 5 = 0 \quad \Rightarrow \quad t = 5
\][/tex]
Therefore, the values of [tex]\( t \)[/tex] that satisfy the equation [tex]\( f(3 - t) = 9 \)[/tex] are [tex]\( t = 1 \)[/tex] and [tex]\( t = 5 \)[/tex].
