Quadratic EquationHard
Question
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 $E_{2}:x^{2} + px + q_{2} = 0$,
Options
A.Nothing can be said about roots of the two equations.
B.atleast one of the equation has distinct real roots.
C.atleast one must have imaginary roots.
D.atleast one must have real roots of opposite sign.
Solution
$\Delta_{1} = 1 - 4q_{1}$
$$\Delta_{2} = p^{2} - 4q_{2}$$
$$\begin{matrix} \Delta_{1} + \Delta_{2} & \ = p^{2} + 1 - 4\left( q_{1} + q_{2} \right) \\ & \ = p^{2} + 1 - 4(p - 1) \\ & \ = (p - 2)^{2} + 1 > 0 \end{matrix}$$
⇒ at least one of $\Delta_{1}$, or $\Delta_{2} > 0$
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