Given,
4 3 x 2 + 5 x − 2 3 = 0 ⇒ 4 3 x 2 + 8 x − 3 x − 2 3 = 0 ⇒ 4 x ( 3 x + 2 ) − 3 ( 3 x + 2 ) = 0 ⇒ ( 3 x + 2 ) ( 4 x − 3 ) = 0 (Factorising left side) ⇒ 3 x + 2 = 0 or 4 x − 3 = 0 (Zero-product rule) ⇒ 3 x = − 2 or 4 x = 3 ⇒ x = − 2 3 or x = 3 4 ⇒ x = − 2 3 × 3 3 or x = 3 4 x = − 2 3 3 or x = 3 4 4\sqrt{3}x^2 + 5x - 2\sqrt{3} = 0 \\[1em] \Rightarrow 4\sqrt{3}x^2 + 8x - 3x - 2\sqrt{3} = 0 \\[1em] \Rightarrow 4x(\sqrt{3}x + 2) - \sqrt{3}(\sqrt{3}x + 2) = 0 \\[1em] \Rightarrow (\sqrt{3}x + 2)(4x - \sqrt{3}) = 0 \text{ (Factorising left side) } \\[1em] \Rightarrow \sqrt{3}x + 2 = 0 \text{ or } 4x - \sqrt{3} = 0 \text{ (Zero-product rule) } \\[1em] \Rightarrow \sqrt{3}x = -2 \text{ or } 4x = \sqrt{3} \\[1em] \Rightarrow x = -\dfrac{2}{\sqrt{3}} \text{ or } x = \dfrac{\sqrt{3}}{4} \\[1em] \Rightarrow x = -\dfrac{2}{\sqrt{3}} \times \dfrac{\sqrt{3}}{\sqrt{3}} \text{ or } x = \dfrac{\sqrt{3}}{4} \\[1em] x = -\dfrac{2\sqrt{3}}{3} \text{ or } x = \dfrac{\sqrt{3}}{4} 4 3 x 2 + 5 x − 2 3 = 0 ⇒ 4 3 x 2 + 8 x − 3 x − 2 3 = 0 ⇒ 4 x ( 3 x + 2 ) − 3 ( 3 x + 2 ) = 0 ⇒ ( 3 x + 2 ) ( 4 x − 3 ) = 0 (Factorising left side) ⇒ 3 x + 2 = 0 or 4 x − 3 = 0 (Zero-product rule) ⇒ 3 x = − 2 or 4 x = 3 ⇒ x = − 3 2 or x = 4 3 ⇒ x = − 3 2 × 3 3 or x = 4 3 x = − 3 2 3 or x = 4 3
Hence, the roots of given equation are − 2 3 3 -\dfrac{2\sqrt{3}}{3} − 3 2 3 3 4 . \dfrac{\sqrt{3}}{4}. 4 3 .
