The given equation is 3 x 2 + 10 x − 8 3 = 0 \sqrt{3}x^2 + 10x - 8\sqrt{3} = 0 3 x 2 + 1 0 x − 8 3 = 0
Comparing it with ax2 + bx + c = 0, we geta = 3 , b = 10 , c = − 8 3 a = \sqrt{3}, \space b =10, \space c = -8\sqrt{3} a = 3 , b = 1 0 , c = − 8 3
By using formula, x = − b ± b 2 − 4 a c 2 a x = \dfrac{-b ± \sqrt{b^2 - 4ac}}{2a} x = 2 a − b ± b 2 − 4 a c
we obtain:
⇒ x = − ( 10 ) ± ( 10 ) 2 − 4 × 3 × − 8 3 2 × 3 ⇒ x = − 10 ± 100 + 96 2 3 ⇒ x = − 10 ± 196 2 3 ⇒ x = − 10 + 14 2 3 or − 10 − 14 2 3 ⇒ x = 4 2 3 or − 24 2 3 ⇒ x = 2 3 or − 12 3 ⇒ x = 2 3 × 3 3 or − 12 3 × 3 3 (Multiplying both roots by 3 3 ) ⇒ x = 2 3 3 or − 4 3 . \Rightarrow x = \dfrac{-(10) ± \sqrt{(10)^2 - 4\times \sqrt{3} \times -8\sqrt{3}}}{2 \times \sqrt{3}} \\[1em] \Rightarrow x = \dfrac{-10 ± \sqrt{100 + 96}}{2\sqrt{3}} \\[1em] \Rightarrow x = \dfrac{-10 ± \sqrt{196}}{2\sqrt{3}} \\[1em] \Rightarrow x = \dfrac{-10 + 14}{2\sqrt{3}} \text{ or } \dfrac{-10 - 14}{2\sqrt{3}}\\[1em] \Rightarrow x = \dfrac{4}{2\sqrt{3}} \text { or } \dfrac{-24}{2\sqrt{3}} \\[1em] \Rightarrow x = \dfrac{2}{\sqrt{3}} \text{ or } \dfrac{-12}{\sqrt{3}} \\[1em] \Rightarrow x = \dfrac{2}{\sqrt{3}} \times \dfrac{\sqrt{3}}{\sqrt{3}} \text{ or } \dfrac{-12}{\sqrt{3}} \times \dfrac{\sqrt{3}}{\sqrt{3}} \text{ (Multiplying both roots by } \dfrac{\sqrt{3}}{\sqrt{3}}) \\[1em] \Rightarrow x = \dfrac{2\sqrt{3}}{3} \text{ or } -4\sqrt{3}. ⇒ x = 2 × 3 − ( 1 0 ) ± ( 1 0 ) 2 − 4 × 3 × − 8 3 ⇒ x = 2 3 − 1 0 ± 1 0 0 + 9 6 ⇒ x = 2 3 − 1 0 ± 1 9 6 ⇒ x = 2 3 − 1 0 + 1 4 or 2 3 − 1 0 − 1 4 ⇒ x = 2 3 4 or 2 3 − 2 4 ⇒ x = 3 2 or 3 − 1 2 ⇒ x = 3 2 × 3 3 or 3 − 1 2 × 3 3 (Multiplying both roots by 3 3 ) ⇒ x = 3 2 3 or − 4 3 .
Hence roots of the given equations are 2 3 3 , − 4 3 \dfrac{2\sqrt{3}}{3} , -4\sqrt{3} 3 2 3 , − 4 3
