If a + b + c = 9 and ab + bc + ca = 26, find a2 + b2 +c2.

Given. $a+b+c=9$ and $a b+b c+c a=26$ ...(i)

Now, $\quad a+b+c=9$

On squaring both sides, we get

$(a+b+c)^{2}=(9)^{2}$

$\Rightarrow \quad a^{2}+b^{2}+c^{2}+2 a b+2 b c+2 c a=81$

[using identity, $\left.(a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2 a b+2 b c+2 c a\right]$

$\Rightarrow \quad a^{2}+b^{2}+c^{2}+2(a b+b c+c a)=81$

$\Rightarrow \quad a^{2}+b^{2}+c^{2}+2(26)=81$    [from Eq. (i)]

$\Rightarrow \quad a^{2}+b^{2}+c^{2}=81-52=29$