Find the zeroes of \(4x^2+5\sqrt{2}x-3\) and verify relations.
\(x=\dfrac{\sqrt{2}}{4},\; x=-\dfrac{3\sqrt{2}}{2}\)
Step 1: Quadratic formula.
\(a = 4, b = 5\sqrt{2}, c = -3\)
Discriminant = \((5\sqrt{2})^2 - 4(4)(-3)\)
= 50 + 48 = 98
\(\sqrt{98} = 7\sqrt{2}\)
Step 2: Roots.
\(x = \dfrac{-5\sqrt{2} + 7\sqrt{2}}{8} = \dfrac{\sqrt{2}}{4}\)
\(x = \dfrac{-5\sqrt{2} - 7\sqrt{2}}{8} = -\dfrac{3\sqrt{2}}{2}\)
Step 3: Verify.
Sum = \(\dfrac{\sqrt{2}}{4} - \dfrac{3\sqrt{2}}{2} = -5\sqrt{2}/4\)
= \(-b/a = -(5\sqrt{2})/4\)
Product = \((\dfrac{\sqrt{2}}{4})(-\dfrac{3\sqrt{2}}{2}) = -3/4\)
= \(c/a = -3/4\)