To find [tex]\(a^2 + b^2 + c^2\)[/tex] given the equations [tex]\(a + b + c = 9\)[/tex] and [tex]\(ab + bc + ca = 26\)[/tex], follow these steps:
1. Start with the identity:
[tex]\[
(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)
\][/tex]
2. Substitute the known values into the identity:
[tex]\[
9^2 = a^2 + b^2 + c^2 + 2 \cdot 26
\][/tex]
3. Simplify the left-hand side:
[tex]\[
81 = a^2 + b^2 + c^2 + 52
\][/tex]
4. Isolate [tex]\(a^2 + b^2 + c^2\)[/tex]:
[tex]\[
a^2 + b^2 + c^2 = 81 - 52
\][/tex]
5. Compute the final result:
[tex]\[
a^2 + b^2 + c^2 = 29
\][/tex]
Hence, [tex]\(a^2 + b^2 + c^2\)[/tex] is [tex]\(29\)[/tex].
