If the quadratic equation  $5{\mathrm{x}}^2-7\mathrm{x}-3=0$ can be expressed as ${(\mathrm{x}-\frac{\mathrm{b}}{\mathrm{c}})}^2 = \frac{109}{100}$, where b and c are co-prime, then what is the value of the expression c² − b²? 

 

Consider the equation: $5{\mathrm{x}}^2-7\mathrm{x}-3=0$

By dividing this equation by 5, we obtain 

${\mathrm{x}}^2-\frac{7}{5}\mathrm{x}-\frac{3}{5}=0$

${\mathrm{x}}^2-2\times \frac{7}{10}\times \mathrm{x}+{\left(\frac{7}{10}\right)}^2-{\left(\frac{7}{10}\right)}^2-\frac{3}{5}=0$

${\left(\mathrm{x}-\frac{7}{10}\right)}^2-\frac{49}{100}-\frac{3}{5}=0$

${\left(\mathrm{x}-\frac{7}{10}\right)}^2-\frac{109}{100}=0$

On comparing this equation with the equation ${\left(\mathrm{x}-\frac{\mathrm{b}}{\mathrm{c}}\right)}^2=\frac{109}{100}$ we obtain b = 7 and c = 10. 

∴ c² − b² = 10² − 7² = 100 − 49 = 51 

Thus, the value of the expression c² − b² is 51.