Question 1196712
<br>
Don't mix upper case and lower case letters when naming your variables. "b" and "B" in the same problem can represent completely different quantities.<br>
We are to evaluate the expression<br>
{{{a^2/b+b^2/a}}}<br>
which, combining the fractions with a common denominator, is equivalent to<br>
{{{(a^3+b^3)/ab}}}<br>
We can work this problem using Vieta's Theorem which says the sum of the roots is {{{a+b=4/3}}} and the product of the roots is {{{ab=-7/3}}}<br>
{{{(a^3+b^3)/ab}}}<br>
Use the factorization pattern for the sum of cubes:<br>
{{{((a+b)(a^2-ab+b^2))/ab}}}<br>
Since we know the values of a+b and ab, rewrite the other term as<br>
{{{a^2-ab+b^2=(a^2+2ab+b^2)-3ab=(a+b)^2-3ab}}}<br>
Then the expression we are to evaluate is all in terms of (a+b) and (ab):<br>
{{{((a+b)((a+b)^2-3ab))/ab}}}<br>
Substitute {{{a+b=4/3}}} and {{{ab=-7/3}}} and evaluate.<br>
{{{((4/3)((4/3)^2-3(-7/3)))/(-7/3)}}}<br>
{{{(-3/7)(((4/3)((4/3)^2-3(-7/3))))}}}<br>
{{{(-4/7)(16/9+7)}}}<br>
{{{-64/63-4}}}<br>
{{{-316/63}}}<br>
ANSWER: {{{a^2/b+b^2/a=-316/63}}}<br>