Given,
⇒ 3 x 2 − 2 x − 1 = 2 x − 2 ⇒ 3 x 2 − 2 x − 1 = ( 2 x − 2 ) 2 (On squaring both sides) ⇒ 3 x 2 − 2 x − 1 = 4 x 2 + 4 − 8 x ⇒ 3 x 2 − 4 x 2 − 2 x + 8 x − 1 − 4 = 0 ⇒ − x 2 + 6 x − 5 = 0 ⇒ x 2 − 6 x + 5 = 0 (On multiplying equation by -1) ⇒ x 2 − 5 x − x + 5 = 0 ⇒ x ( x − 5 ) − 1 ( x − 5 ) = 0 ⇒ ( x − 1 ) ( x − 5 ) = 0 ⇒ x − 1 = 0 or x − 5 = 0 x = 1 or x = 5 \Rightarrow \sqrt{3x^2 - 2x - 1} = 2x - 2 \\[1em] \Rightarrow 3x^2 - 2x - 1 = (2x - 2)^2 \text{ (On squaring both sides) } \\[1em] \Rightarrow 3x^2 - 2x - 1 = 4x^2 + 4 - 8x \\[1em] \Rightarrow 3x^2 - 4x^2 - 2x + 8x - 1 - 4 = 0 \\[1em] \Rightarrow -x^2 + 6x - 5 = 0 \\[1em] \Rightarrow x^2 - 6x + 5 = 0 \text{ (On multiplying equation by -1) }\\[1em] \Rightarrow x^2 - 5x - x + 5 = 0 \\[1em] \Rightarrow x(x - 5) - 1(x - 5) = 0 \\[1em] \Rightarrow (x - 1)(x - 5) = 0 \\[1em] \Rightarrow x - 1 = 0 \text{ or } x - 5 = 0 \\[1em] x = 1 \text{ or } x = 5 ⇒ 3 x 2 − 2 x − 1 = 2 x − 2 ⇒ 3 x 2 − 2 x − 1 = ( 2 x − 2 ) 2 (On squaring both sides) ⇒ 3 x 2 − 2 x − 1 = 4 x 2 + 4 − 8 x ⇒ 3 x 2 − 4 x 2 − 2 x + 8 x − 1 − 4 = 0 ⇒ − x 2 + 6 x − 5 = 0 ⇒ x 2 − 6 x + 5 = 0 (On multiplying equation by -1) ⇒ x 2 − 5 x − x + 5 = 0 ⇒ x ( x − 5 ) − 1 ( x − 5 ) = 0 ⇒ ( x − 1 ) ( x − 5 ) = 0 ⇒ x − 1 = 0 or x − 5 = 0 x = 1 or x = 5
Since we squared the equation, so roots need to be checked
Putting x = 1 in equation
⇒ 3 ( 1 ) 2 − 2 ( 1 ) − 1 = 2 ( 1 ) − 2 ⇒ 0 = 0 \Rightarrow \sqrt{3(1)^2 - 2(1) - 1} = 2(1) - 2 \\[1em] \Rightarrow \sqrt{ 0 } = 0 \\[1em] ⇒ 3 ( 1 ) 2 − 2 ( 1 ) − 1 = 2 ( 1 ) − 2 ⇒ 0 = 0
L.H.S. = R.H.S. = 0
Putting x = 5 in equation
⇒ 3 ( 5 ) 2 − 2 ( 5 ) − 1 = 2 ( 5 ) − 2 ⇒ 64 = 8 \Rightarrow \sqrt{3(5)^2 - 2(5) - 1} = 2(5) - 2 \\[1em] \Rightarrow \sqrt{ 64 } = 8 \\[1em] ⇒ 3 ( 5 ) 2 − 2 ( 5 ) − 1 = 2 ( 5 ) − 2 ⇒ 6 4 = 8
L.H.S. = R.H.S. = 8
Hence, roots of the equation are 1, 5.
