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The roots of x^2 + ax + b = 0 are a and b.  Find all possible pairs (a,b).

 May 14, 2020
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Since you know that the roots are  a  and  b, the quadratic can be written as  (x - a)(x - b)  =  0

 

(x - a)(x - b)  =  0     --->     x2 - ax - bx + ab  =  0     --->     x2 + (-a -b)x + ab  =  0

 

We are given that the equation is:  x2 + ax + b  =  0

 

Comparing these two equations, we see that  a  =  -a - b     and     b  =  ab

                                                            

If b  =  ab,  a  must be 1.

 

If  a  =  -a - b     --->     2a  =  - b     --->     b =  -2a

 

Since  a = 1     --->     b  =  -2

 

I believe that there is only one ordered pair,  (1, -2)

 May 14, 2020

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