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Least value of a sum of squares of real roots

Least value of a sum of squares of real roots

IP 3
8 marks

The equation \(x^2+(k-3)x+k=0\), where \(k\) is a real constant, has real roots \(\alpha\) and \(\beta\).

  1. Find the set of possible values of \(k\).[3]
  2. Show that \(\alpha^2+\beta^2=k^2-8k+9\).[2]
  3. Hence find the least possible value of \(\alpha^2+\beta^2\), justifying your answer.[3]

Solution:

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Answer:(a) \(k\le 1\) or \(k\ge 9\) (b) shown (c) \(2\), when \(k=1\)

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