Find the zeroes of \(y^2+\dfrac{\sqrt{3}}{2}y-5\) and verify relations.
\(y=\dfrac{-\sqrt{3}+\sqrt{83}}{4},\; y=\dfrac{-\sqrt{3}-\sqrt{83}}{4}\)
Step 1: Identify coefficients.
\(a=1, b=\sqrt{3}/2, c=-5\)
Step 2: Discriminant.
\(b^2 - 4ac = (\sqrt{3}/2)^2 - 4(1)(-5)\)
= 3/4 + 20 = 83/4
\(\sqrt{83/4} = \sqrt{83}/2\)
Step 3: Roots.
\(y = \dfrac{-\sqrt{3}/2 ± \sqrt{83}/2}{2}\)
= \(\dfrac{-\sqrt{3} ± \sqrt{83}}{4}\)
Step 4: Verify.
Sum = \((-\sqrt{3} + \sqrt{83})/4 + (-\sqrt{3} - \sqrt{83})/4\)
= -2\sqrt{3}/4 = -\sqrt{3}/2 = -b/a
Product = \((-\sqrt{3} + \sqrt{83})(-\sqrt{3} - \sqrt{83})/16\)
= (83 - 3)/16 = 80/16 = 5 = c/a