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(C) Let the roots of the quadratic equation $x^2 - px + q = 0$ be $\alpha$ and $\beta$,where $\alpha, \beta \in \mathbb{Z}^+$.By the properties of roo

Vedclass · Accepted Answer

$1, 2$

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(C) Let the roots of the quadratic equation $x^2 - px + q = 0$ be $\alpha$ and $\beta$,where $\alpha, \beta \in \mathbb{Z}^+$.By the properties of roots,we have:$\alpha + \beta = p$ (sum of roots)$\alpha \cdot \beta = q$ (product of roots)Since $q$ is a prime number,its only factors are $1$ and $q$. Thus,the roots must be $1$ and $q$.Substituting these into the sum of roots equation: $1 + q = p$.Since $p$ and $q$ are both prime numbers,we look for two primes that differ by $1$. The only such primes are $2$ and $3$ (where $q=2$ and $p=3$).Substituting $p=3$ and $q=2$ into the equation: $x^2 - 3x + 2 = 0$.Factoring the quadratic: $(x - 1)(x - 2) = 0$.Thus,the roots are $1$ and $2$.

If $p$ and $q$ are distinct prime numbers and the equation $x^2 - px + q = 0$ has positive integers as its roots,then the roots of the equation are:

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