Sure, let's walk through the solution step-by-step.
Given:
[tex]\[ a - b = 8 \][/tex]
[tex]\[ ab = 10 \][/tex]
We need to find [tex]\( a^2 + b^2 \)[/tex].
We can use the identity for the square of a binomial and express [tex]\( a^2 + b^2 \)[/tex] in terms of [tex]\( (a - b)^2 \)[/tex] and the product [tex]\( ab \)[/tex]:
First, let's recall the algebraic identity:
[tex]\[ (a - b)^2 = a^2 - 2ab + b^2 \][/tex]
This can be rearranged to express [tex]\( a^2 + b^2 \)[/tex]:
[tex]\[ a^2 + b^2 = (a - b)^2 + 2ab \][/tex]
We already know from the given conditions:
[tex]\[ a - b = 8 \][/tex]
[tex]\[ ab = 10 \][/tex]
First, calculate [tex]\( (a - b)^2 \)[/tex]:
[tex]\[ (a - b)^2 = 8^2 = 64 \][/tex]
Next, we need to find [tex]\( 2ab \)[/tex]:
[tex]\[ 2ab = 2 \times 10 = 20 \][/tex]
Now, adding these results together:
[tex]\[ a^2 + b^2 = (a - b)^2 + 2ab = 64 + 20 = 84 \][/tex]
Therefore, the value of [tex]\( a^2 + b^2 \)[/tex] is:
[tex]\[ \boxed{84} \][/tex]
