Question:
If \[ a+b=10 \quad \text{and} \quad ab=16 \] find the values of:
\[ a^2-ab+b^2 \] and \[ a^2+ab+b^2 \]
Solution:
Using identity:
\[ (a+b)^2=a^2+2ab+b^2 \]
\[ 10^2=a^2+2(16)+b^2 \]
\[ 100=a^2+32+b^2 \]
\[ a^2+b^2=100-32 \]
\[ a^2+b^2=68 \]
Now,
\[ a^2-ab+b^2 \]
\[ =68-16 \]
\[ =52 \]
Also,
\[ a^2+ab+b^2 \]
\[ =68+16 \]
\[ =84 \]