Find the zeroes of $4 \sqrt{3} x^{2} + 5 x - 2 \sqrt{3}$ and verify the relation between the zeroes and coefficient of the polynomial.

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Solution

$4 \sqrt{3} x^{2} + 5 x - 2 \sqrt{3}$

$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

$= \frac{- 5 \pm \sqrt{(5)^{2} - 4 (4 \sqrt{3}) (- 2 \sqrt{3})}}{2 (4 \sqrt{3})}$

$= \frac{- 5 \pm \sqrt{25 + 96}}{8 \sqrt{3}} = \frac{- 5 \pm 11}{8 \sqrt{3}}$

$α = \frac{- 16}{8 \sqrt{3}}$ & $β = \frac{6}{8 \sqrt{3}}$

$α = \frac{- 2}{\sqrt{3}}$ & $β = \frac{3}{4 \sqrt{3}}$

$α + β = \frac{- 2}{\sqrt{3}} + \frac{3}{4 \sqrt{3}} = \frac{- 5}{\sqrt{3}}$

$α + β = \frac{- b}{a} = \frac{- 5}{4 \sqrt{3}}$

$α β = (\frac{- 2}{\sqrt{3}}) (\frac{3}{4 \sqrt{3}}) = \frac{- 1}{2}$

$α β = \frac{c}{a} = \frac{- 2 \sqrt{3}}{4 \sqrt{3}} = \frac{- 1}{2}$ .

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