If $p,q_{1}$ and $q_{2}$ are real numbers with $p = q_{1} + q_{2} + 1$, then which of the following

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If $p,q_{1}$ and $q_{2}$ are real numbers with $p = q_{1} + q_{2} + 1$, then which of the following must be correct about the equations $E_{1}:x^{2} + x + q_{1} = 0$ and $E_{2}:x^{2} + px + q_{2} = 0$,

Accepted Answer

$\Delta_{1} = 1 - 4q_{1}$$$\Delta_{2} = p^{2} - 4q_{2}$$$$\begin{matrix} \Delta_{1} + \Delta_{2} & \ = p^{2} + 1 - 4\left( q_{1} + q_{2} ight) \ & \ = p^{2} + 1 - 4(p - 1) \ & \ = (p - 2)^{2} + 1 > 0 \end{matrix}$$⇒ at least one of $\Delta_{1}$, or $\Delta_{2} > 0$

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