Consider the cubic equation x3+px2+qx+r=0, where p, q, r are real numbers. Which of the following statements is correct?
A
If p2−2q<0, then the equation has one real and two imaginary root.
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B
If p2−2q≥0, then the equation has all real roots.
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C
If p2−2q>0, then the equation has all real and distinct roots.
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D
If 4p3−27q2>0, then the equation has real and distinct roots.
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Solution
The correct option is A If p2−2q<0, then the equation has one real and two imaginary root. Let f(x)=x3+px2+qx+r ∴f′(x)=3x2+2px+q Disc =4p2−12q=4(p2−3q) =4(p2−2q−q) ∴ If p2<2q⇒p2<3q So, the equation f(x)=0 has one real and two imaginary roots.