Let \( p \) and \( q \) be real numbers such that \( x^{2}+p x+q \neq 0 \) for every real number...

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Let \( p \) and \( q \) be real numbers such that \( x^{2}+p x+q \neq 0 \) for every real number \( x \). Prove that if \( n \) is an odd positive integer, then \( X^{2}+p X+q I_{n} \neq 0_{n} \) for all real matrices \( X \) of order \( n \times n \).
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