Given,
32x2−31x=1⇒32x2−31x−1=0 (Writing as ax2+bx+c=0)⇒32x2×3−31x×3−1×3=0×3 (Multiplying the equation by 3) ⇒2x2−x−3=0⇒2x2−3x+2x−3=0⇒x(2x−3)+1(2x−3)=0⇒(x+1)(2x−3)=0 (Factorising left side) ⇒x+1=0 or 2x−3=0 (Zero-product rule) ⇒x=−1 or x=23
Hence, the roots of given equation are -1, 23.