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Let $a$ and $b$ be the roots of the quadratic equation $x^2-25x+80=-28x+75$. Compute $\frac{a^2}{b} + \frac{b^2}{a}$.

 Sep 8, 2023
 #1
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\(\frac{a^2}{b} + \frac{b^2}{a} \)

 

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

 

cool cool cool

 Sep 8, 2023

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