+0  
 
0
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(Since A  is representing a digit, if A = 2 then A0 represents the integer 20.)

 Jan 28, 2021
 #1
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+1

x^2 -wx - 10w = 0       (I chnged a to w for clarity)

Discriminant must be > 0

 

b^2 - 4ac > 0

w^2 -4(1)(-10w) >0

w^2 + 40w >0

w(w + 40) > 0       both w and (w+40)  must be positive for this to be true

 

w >0   and    w >-40      is restricted to    w > 0        1 2 3 4 5 6 7 8 9    (since it must be positive single digit per Question)

 Jan 28, 2021
 #2
avatar+116126 
+1

Let 1A   =    10  +  A

Let  A0   =    10A

 

So.....if we  have  integer  solutions, the  discriminant  must  be a  perfect square   

 

So

 

( 10+A)^2   -  4 (10A)       =

 

100 + 20A  + A^2  -  40A   = 

 

A^2  - 20A  +  100     =

 

(A - 10) ( A  -10)   = 

 

(A   - 10)^2

 

This  will   be a perfect square for any integer    A  >   0

 

But since A  must  be a single digit....then....

 

   A   can  be any positive  integer from   1  -  9   inclusive

 

cool cool cool

 Jan 28, 2021

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