Let $a$ and $b$ be the roots of the quadratic equation $x^2-25x+80=-28x+75$. Compute $\frac{a^2}{b} + \frac{b^2}{a}$.
+129881 \(\frac{a^2}{b} + \frac{b^2}{a} \)
Note that the above can be simplified as
[ a^3 + b^3 ] / [ ab ] = [ (a + b) (a^2 - ab + b^2) ] / [ ab] = [ (a + b) (a^2 + b^2 - ab) ] / [ab]
Simplify the equation as
x^2 + 3x + 5 = 0
The sum of the roots, a + b = -3 square both sides
a^2 + 2ab + b^2 = 9 (2)
The product of the roots = ab = 5
2ab = 10
Sub this into (2)
a^2 + 10 + b^2 = 9
a^2 + b^2 = -1
Sub all this into (1)
[ (-3) ( -1 -5) ] / [5] =
[ (-3)(-6) ] / [5]
18 / 5
