To find the value of [tex]\( t \)[/tex] given the equation
[tex]\[
k = \frac{a^2 + t}{2 - a}
\][/tex]
where [tex]\( a = 4 \)[/tex] and [tex]\( k = -11 \)[/tex], we will follow these steps:
1. Substitute the given values into the equation:
[tex]\[
-11 = \frac{4^2 + t}{2 - 4}
\][/tex]
2. Evaluate the expressions inside the fraction:
[tex]\[
-11 = \frac{16 + t}{-2}
\][/tex]
3. Simplify the right-hand side of the equation:
[tex]\[
-11 = \frac{16 + t}{-2}
\][/tex]
To eliminate the fraction, multiply both sides of the equation by -2:
[tex]\[
(-11) \times (-2) = 16 + t
\][/tex]
[tex]\[
22 = 16 + t
\][/tex]
4. Solve for [tex]\( t \)[/tex]:
Subtract 16 from both sides of the equation:
[tex]\[
t = 22 - 16
\][/tex]
[tex]\[
t = 6
\][/tex]
Therefore, the value of [tex]\( t \)[/tex] is:
[tex]\[
\boxed{6}
\][/tex]
