A root of the equation $( x - 1 ) ( x - 2 ) = \dfrac { 30 } { 49 }$ is
- $\dfrac { 17 } { 7 }$
- $\dfrac { 15 } { 7 }$
- $\dfrac { 13 } { 7 }$
- $\dfrac { 11 } { 7 }$
(x - 1)(x - 2) = 30/49. x^2 - 3x + 2 - 30/49 = 0. x^2 - 3x + 68/49 = 0. Using quadratic formula: x = [3 +/- sqrt(9 - 4*68/49)] / 2 = [3 +/- sqrt(441/49 - 272/49)] / 2 = [3 +/- sqrt(169/49)] / 2 = [3 +/- 13/7] / 2. Roots are (21/7 + 13/7)/2 = 34/14 = 17/7 and (21/7 - 13/7)/2 = 8/14 = 4/7.
Expanding the left side gives x^2 - 3x + 2 = 30/49. Multiplying by 49 and moving all terms to the left gives 49x^2 - 147x + 68 = 0. Using the quadratic formula, x = (147 +- sqrt(147^2 - 4*49*68)) / (2*49). The discriminant is 9409, which is 97^2, resulting in x = (147 +- 97) / 98, which gives the roots as 244/98 and 50/98. Simplifying these fractions gives 17/7, which is the required root.