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 $\Delta_{2} > 0$
Create a free account to view solution
View Solution FreeMore Quadratic Equation Questions
The factors of 2x2 − x + p are rational if -...If α and β are roots of x2 − 2x + 3 = 0, then the equation whose roots are and will be -...If the roots of both the equations px2 + 2qx + r = 0 and qx2 − 2x + q = 0 are real, then -...If α, β are roots of the equation (3x + 2)2 + p ( 3x + 2) + q = 0, then roots of x2 + px + q = 0 are -...If $ax^{2} + bx + c = 0,a \neq 0$ be an equation with integral coefficients and $D > 0$ be its discriminant. Then the...