Consider the equation ${x^2} + \alpha x + \beta = 0$ having roots $\alpha ,\beta $ such that $\alpha \ne \beta $ .Also consider the inequality $\left| {\left| {y - \beta } \right| - \alpha } \right| < \alpha $ ,then
inequality is satisfied by exactly two integral values of $y$
inequality is satisfied by all values of $y \in (-4, 2)$
Roots of the equation are of same sign
${x^2} + \alpha x + \beta > 0\,\forall \,x \in \,\left[ { - 1,0} \right]$
The number of real roots of the equation $x | x |-5| x +2|+6=0$, is
Let $\alpha, \beta ; \alpha>\beta$, be the roots of the equation $x^2-\sqrt{2} x-\sqrt{3}=0$. Let $P_n=\alpha^n-\beta^n, n \in N$. Then $(11 \sqrt{3}-10 \sqrt{2}) \mathrm{P}_{10}+(11 \sqrt{2}+10) \mathrm{P}_{11}-11 \mathrm{P}_{12}$ is equal to :
The least integral value $\alpha $ of $x$ such that $\frac{{x - 5}}{{{x^2} + 5x - 14}} > 0$ , satisfies
Let $x_1, x_2, \ldots, x_6$ be the roots of the polynomial equation $x^6+2 x^5+4 x^4+8 x^3+16 x^2+32 x+64=0$. Then,
The number of real solutions of the equation $|{x^2} + 4x + 3| + 2x + 5 = 0 $are