darshshah981 wrote:
If \(a1\) and \(a2\) are the real roots of \(x^2-px+12=0\), then which of the following statements is definitely true?
a) |\(a1+a2\)| \(\leq\) 2\(\sqrt{3}\)
b) |\(a1-a2\)| \(\leq\) 2\(\sqrt{3}\)
c) |\(a1+a2\)| \(\geq\) 4\(\sqrt{3}\)
d) |\(a1-a2\)| \(\geq\) 4\(\sqrt{3}\)
e) None of the above
\(x^2-px+12=0\)
Lets look at this quadratic equation is the form of \(ax^2+bx+c=0\)
We know that sum of the roots of a quadratic is \(-b/a\) and the product of the roots of a quadratic is \(c/a\)
Now, if the roots are real then the discriminant \(\geq\) 0, that is \(b^2 - 4ac\) \(\geq\) 0
>> \(p^2 - 4(1)(12)\) \(\geq\) 0
>> \(p^2\) \(\geq\) 48
>> |\(p\)| \(\geq\) \(\sqrt{48}\)
>> |\(p\)| \(\geq\) \(4\sqrt{3}\)
But p is the sum of the roots (a1+a2)...(Since sum of the roots of a quadratic equation is \(-b/a\))
Therefore, |\(a1+a2\)| \(\geq\) 4\(\sqrt{3}\) or C is the correct answer.
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