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Question

Find rational numbers a and b such that

(i) 2-12+1=a+b2

(ii) 2-52+5=a5+b

(iii) 3+23-2=a+b6

(iv) 5+237+43=a+b3

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Solution


(i)
2-12+1=2-12+1×2-12-1=2-1222-12
=2+1-222-1=3-22
2-12+1=3+-22=a+b2a=3, b=-2

(ii)
2-52+5=2-52+5×2-52-5=2-5222-52
=4+5-454-5=9-45-1=-9+45
2-52+5=45+-9=a5+ba=4, b=-9

(iii)
3+23-2=3+23-2×3+23+2=3+2232-22
=3+2+2×3×23-2=5+26
3+23-2=5+26=a+b6a=5, b=2

(iv)
5+237+43=5+237+43×7-437-43=35-203+143-2472-432
=11-6349-48=11-63
5+237+43=11+-63=a+b3a=11,b=-6

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