If the quadratic equation 5x² − 7x − 3 = 0 can be expressed as ${\left(x-\frac{b}{c}\right)}^2=\frac{109}{100}$, where b and c are co-prime, then what is the value of the expression c² − b²?

Given: 5x² − 7x − 3 = 0

Dividing the given equation by 5, we obtain

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

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

$\Rightarrow {\left(x-\frac{7}{10}\right)}^2=\frac{49}{100}+\frac{3}{5}=\frac{109}{100}$

Comparing this equation with the equation ${\left(x-\frac{b}{c}\right)}^2=\frac{109}{100}$, we obtain

b = 7 and c = 10

∴c² − b² = 10² − 7² = 100 − 49 = 51

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