Find the zeroes of \(2s^2-(1+2\sqrt{2})s+\sqrt{2}\) and verify relations.
\(s=\dfrac{1}{2},\; s=\sqrt{2}\)
Step 1: Check zeroes.
Substitute \(s = \sqrt{2}\)
\(2(2) - (1 + 2\sqrt{2})(\sqrt{2}) + \sqrt{2}\)
= 4 - \sqrt{2} - 4 + \sqrt{2} = 0
So, \(s = \sqrt{2}\) is a root.
Substitute \(s = 1/2\)
\(2(1/4) - (1 + 2\sqrt{2})(1/2) + \sqrt{2}\)
= 1/2 - 1/2 - \sqrt{2} + \sqrt{2} = 0
So, \(s = 1/2\) is a root.
Step 2: Verify.
Sum = \(1/2 + \sqrt{2}\)
= \((1 + 2\sqrt{2})/2 = -b/a\)
Product = \((1/2)(\sqrt{2}) = \sqrt{2}/2\)
= c/a