Certainly! Let's solve this step-by-step.
Given:
[tex]\[ a + b = 9 \][/tex]
[tex]\[ ab = 4 \][/tex]
We need to find the value of [tex]\( a^2 + b^2 \)[/tex].
First, let's use the square of the sum of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[ (a + b)^2 = a^2 + b^2 + 2ab \][/tex]
We have [tex]\( a + b = 9 \)[/tex], so:
[tex]\[ (a + b)^2 = 9^2 = 81 \][/tex]
Now substitute [tex]\( (a + b)^2 = 81 \)[/tex] and [tex]\( ab = 4 \)[/tex] into the equation:
[tex]\[ 81 = a^2 + b^2 + 2 \cdot 4 \][/tex]
Simplify the equation:
[tex]\[ 81 = a^2 + b^2 + 8 \][/tex]
Subtract 8 from both sides to isolate [tex]\( a^2 + b^2 \)[/tex]:
[tex]\[ 81 - 8 = a^2 + b^2 \][/tex]
[tex]\[ 73 = a^2 + b^2 \][/tex]
Thus, the value of [tex]\( a^2 + b^2 \)[/tex] is:
[tex]\[ \boxed{73} \][/tex]
