In the adjoining figure, ABCD is …

We know that the diagonals of a trapezium divide each other proportionally.
So, {tex}\frac { A O } { O C } = \frac { D O } { O B }{/tex}
{tex}\Rightarrow \frac { 5 x - 7 } { 2 x + 1 } = \frac { 7 x - 5 } { 7 x + 1 }{/tex}

By cross multiplication method we have
{tex} 35 x ^ { 2 } - 49 x + 5 x - 7{/tex}{tex}= 14 x ^ { 2 } + 7 x - 10 x - 5{/tex}
{tex}\Rightarrow 21 x ^ { 2 } - 41 x - 2 = 0{/tex}

Factorise the given quadratic equation we have
    {tex} 21 x ^ { 2 } - 42 x + x - 2 = 0{/tex}
{tex}\Rightarrow 21 x ( x - 2 ) + 1 ( x - 2 ) = 0{/tex}
{tex}\Rightarrow ( 21 x + 1 ) ( x - 2 ) = 0{/tex}
Either {tex} x - 2 = 0 {/tex} or {tex}21 x + 1 = 0{/tex}
{tex}\Rightarrow x = 2 {/tex} or {tex}x = \frac { - 1 } { 21 }{/tex}
{tex}x = \frac { - 1 } { 21 } {/tex} is Rejected.
Therefore {tex}x = 2{/tex}