👉 The statement "the sum of the squares of the roots of a quadratic equation is equal to twice the product of its roots" contains an invalid mathematical premise. This is not generally true for all quadratic equations. For a quadratic equation in the form \( ax^2 + bx + c = 0 \), the sum of the squares of its roots (\( r_1 \) and \( r_2 \)) is not equal to twice the product of its roots (\( r_1r_2 \)). The correct relationship is given by Vieta's formulas: the sum of the roots \( r_1 + r_2 = -\frac{b}{a} \) and the product of the roots \( r_1r_2 = \frac{c}{a} \). Squaring the sum of the roots gives \( (r_1 + r_2)^2 = r_1^2 + 2r_1r_2 + r_2^2 \), but this does not equal \( 2r_1r_2 \) unless the quadratic equation is specifically structured in a way that enforces this relationship, which is not universally applicable. Thus, the claim is mathematically incorrect for general quadratic equations.
