High School Challenge 3

Algebra Level 3

In the equation \((x-y)(x-z)\), \(y\) and \(z\) are the shorter sides of a right-angled triangle where \(x\) is a real number and the hypotenuse is of length \(\sqrt{b^{2}+2bc}\) . If \((x-y)(x-z) = x^{2}+bx+c\) (where b and c are real numbers), find \(\frac{y(y-1)}{z(z-1)}\).

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