Certainly! Let's solve this step-by-step.
We are given the two equations:
1. [tex]\( a + b + c = 9 \)[/tex]
2. [tex]\( ab + bc + ca = 40 \)[/tex]
We will use the identity:
[tex]\[ (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca) \][/tex]
First, substitute [tex]\( a + b + c = 9 \)[/tex] and [tex]\( ab + bc + ca = 40 \)[/tex] into the identity.
The left-hand side of the identity is:
[tex]\[ (a + b + c)^2 = 9^2 = 81 \][/tex]
We also know that:
[tex]\[ 2(ab + bc + ca) = 2 \times 40 = 80 \][/tex]
Now, we substitute these values into the identity:
[tex]\[ 81 = a^2 + b^2 + c^2 + 80 \][/tex]
To isolate [tex]\( a^2 + b^2 + c^2 \)[/tex], subtract 80 from both sides:
[tex]\[ 81 - 80 = a^2 + b^2 + c^2 \][/tex]
Thus:
[tex]\[ a^2 + b^2 + c^2 = 1 \][/tex]
So, the value of [tex]\( a^2 + b^2 + c^2 \)[/tex] is [tex]\(\boxed{1}\)[/tex].
