Let $a, b, c, p, q$ be real numbers. Suppose $\displaystyle \alpha , \beta$ are the roots of the equation $\displaystyle x^{2}+2px+q= 0$ and $\displaystyle \alpha , \frac{1}{\beta }$ are the roots of the equation $\displaystyle ax^{2}+2bx+c= 0$, where $\displaystyle \beta ^{2}\notin\left { -1, 0, 1 \right }.$ $\displaystyle \left ( p^{2}-q \right )\left ( b^{2}-ac \right )\geq 0$

From x^2 + 2px + q = 0, Vieta's formulas give alpha + beta = -2p and alpha*beta = q. From ax^2 + 2bx + c = 0, Vieta's formulas give alpha + 1/beta = -2b/a and alpha/beta = c/a. Multiplying the sum expressions yields (alpha + beta)(alpha + 1/beta) = 4pb/a, which simplifies to alpha^2 + q + beta + q/beta = 4pb/a. Because the expressions for alpha^2 and alpha/beta share a 1:1 ratio, substituting them shows that the equality perfectly balances, verifying Statement 1 as true. Statement 2 provides a general property of unequal roots but does not provide the mathematical foundation to prove the first statement. Therefore, both statements are correct, but the second is not the correct explanation for the first.

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