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Given that the absolute value of the difference of the two roots of ax^2 + 5x - 3 = 0 is \(7/2\), and a is positive, what is the value of a?

 Apr 29, 2022
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Let the roots  be m and n

The sum of the roots  =  -5/a  =  m + n

Square both sides of this

25/a^2  = m^2  + 2mn + n^2

 

The product of the roots =  - 3/a = mn

Then    -4(-3)/a  =  -4mn

So      12/a  =  -4mn

 

So

 

25/a^2  + 12/a   =    m^2 + 2mn - 4mn  + n^2

 

25/a^2  + 12/a  =  m^2 - 2mn  + n^2

 

25/a^2  + 12/a  = (m +n) (m - n)

 

25/a^2 + 12/a  =  (-5/a)(m - n)        multiply through by   a

 

25/a  + 12  = -5 ( m - n)

 

If we let   m - n  =   -7/2     then we have that

 

25/a  + 12  = -5(-7/2)

 

25/a  + 12  =  35/2             multiply through by 2

 

50/a + 24  =  35

 

50 / a  =   21

 

a  =  50  / 21

 

 

cool cool cool

 Apr 30, 2022

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