A) imaginary
B) real roots
C) purely imaginary roots
D) none of these
Correct Answer: B
Solution :
[b]\[Let\,\,{{D}_{1}}\,\,and\text{ }{{D}_{2}}\] be desicriminants of \[{{x}^{2}}+{{b}_{1}}x+{{c}_{1}}=0\] |
\[and\text{ }{{x}^{2}}+{{b}_{2}}x+{{c}_{2}}=0,respectively.\,\,Then,\] |
\[{{D}_{1}}+{{D}_{2}}={{b}_{1}}^{2}-4{{c}_{1}}+{{b}_{2}}^{2}-4{{c}_{2}}\] |
\[=({{b}_{1}}^{2}+{{b}_{2}}^{2})-4({{c}_{1}}+{{c}_{2}})\] |
\[={{b}_{1}}^{2}+{{b}_{2}}^{2}-2{{b}_{1}}{{b}_{2}}[\therefore {{b}_{1}}{{b}_{2}}=2({{c}_{1}}+{{c}_{2}})]\] |
\[={{({{b}_{1}}-{{b}_{2}})}^{2}}\ge 0\] |
\[\Rightarrow {{D}_{1}}\ge 0\,\,or\text{ }{{D}_{2}}\ge 0\,\,or\text{ }{{D}_{1}}\,and\,{{D}_{2}}\] both are positive |
Hence, at least one of the equations has real roots. |
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