Positive geometric roots of a cubic
Positive geometric roots of a cubic
IB Year 6 | Grade 12
8 marks
The three positive roots of \(x^3-10x^2+qx-27=0\) are consecutive terms of an increasing geometric sequence.
- Show that the middle term is \(3\).[2]
- Find \(q\).[2]
- Find all three roots and the common ratio, giving exact values.[4]
Solution:
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Answer:(a) \(3\) (b) \(q=30\) (c) \(\frac{7-\sqrt{13}}{2},\ 3,\ \frac{7+\sqrt{13}}{2}\) and \(r=\frac{7+\sqrt{13}}{6}\)
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