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Let X= -5,-4,-3,-2,-1,0,1,2,3,4,5 and S= (a, b) X X: x^2+a x+b . and x^3+b x+a have at least a common real zero . How many elements are there in S ?

Correct answer

24

Step-by-step solution

Let be a common root, then aligned & ^2+a +b=0 ...(i) & ^3+b +a=0 ...(ii) aligned (i) - (ii) gives, a ( ^2-1 )=0 a=0 or =1 or =-1 Case 1: When a=0 then b 0 Total 6 cases. Case II: When =1 then a+b=-1 a=-5,-4,-3,-2,-1,1,2,3,4 (9 cases) Case III: When =-1 then a-b=1 a=5,4,3,2,1,-1,-2,-3,-4 (9 cases) There exist total 24 ordered pairs (a, b)

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