A) \[4{{a}^{2}}c-{{b}^{2}}=8{{a}^{3}}f\]
B) \[4{{a}^{2}}c=8{{a}^{3}}f\]
C) \[2{{a}^{3}}c={{a}^{3}}f\]
D) none of these
Correct Answer: A
Solution :
We have,\[{{a}^{2}}{{x}^{4}}+b{{x}^{3}}+c{{x}^{2}}+dx+{{f}^{2}}\] \[={{(a{{x}^{2}}+cx+f)}^{2}}a\]a perfect square \[={{a}^{2}}{{x}^{4}}+2ac{{x}^{3}}+(2af+{{c}^{2}}){{x}^{2}}+2cfx+{{f}^{2}}\] \[\therefore \]\[b=2ac,\,\,c=2af+{{c}^{2}},\,\,d=2cf\]and Again \[4{{a}^{2}}c=4{{a}^{2}}(2af+{{c}^{2}})=8{{a}^{3}}f+{{b}^{2}}\] \[(\because \,\,b=2ac)\] \[\therefore \] \[4{{a}^{2}}c={{b}^{2}}+8{{a}^{3}}f\]
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