+1725 Let $a$ and $b$ be the roots of the quadratic equation $x^2-25x+80=-28x+75$. Compute $\frac{a^2}{b} + \frac{b^2}{a}$.
+129771 a^2 / b + b^2 / a = [ a^3 + b^3 ] / [ab ] ^2 = [(a + b) (a^2 + b^2 - ab ) ] / [ab]
x^2 -25x + 80 = -28x + 75 simplify
x^2 + 3x + 5 = 0
We have the form mx^2 + nx + c = 0
Sum of roots
a + b = -n/m = -3/1 = -3
Product of roots = ab = c / m = 5/1 = 5
a + b = -3 square both sides
a^2 + 2ab + b^2 = 9
a^2 +2(5) + b^2 = 9
a^2 + 10 + b^2 = 9
a^2 + b^2 = -1
So
[(a + b) (a^2 + b^2 - ab) ] / [ab] =
[ ( -3)( -1 - 5) ] / [5] =
[ 18 ] / [5 ] =
18 / 5 = 3.6
