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

 Aug 24, 2023
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avatar+128393 
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x^2-25x+80 = -28x+75

 

x^2 + 3x + 5  = 0

 

Note that       a/b  +  b/a =   [a^2 + b^2 ] / [ab ]

 

Sum of roots = a + b =  -3     square both sides   

a^2 + 2ab + b^2  =  9        (1)

 

Product of roots = 5

So  ab = 5

2ab = 10       (2)

 

Sub (2) into (1)

 

a^2 +  10 + b^2    =  9

a^2 + b^2  =   -1

 

So  a/b + b/a =   -1 / 5

 

cool cool cool

 Aug 24, 2023

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