Question
The real roots of the equation $${x^2} + 5\left| x \right| + 4 = 0$$ are
A.
$$\left\{ { - 1, - 4} \right\}$$
B.
$$\left\{ { 1, 4} \right\}$$
C.
$$\left\{ { - 4, 4} \right\}$$
D.
None of these
Answer :
None of these
Solution :
Case 1 : $$x \geqslant 0$$
∴ the equation becomes $$x^2 + 5x + 4 =0$$ or $$x = - 1, - 4$$ but $$x \geqslant 0$$
∴ both values, non admissible :
Case 2 : $$x \leqslant 0$$
The equation becomes $$x^2 - 5x + 4 = 0$$ or $$x = 1, 4$$ both values are non admissible
∴ No real roots.
Alternatively, since $${x^2} \geqslant 0;\left| x \right| \geqslant 0$$
$$\eqalign{
& \therefore {x^2} + \left| x \right| + 4 > 0{\text{ for all }}x \in {\bf{R}} \cr
& \therefore {x^2} + \left| x \right| + 4 \ne 0{\text{ for any }}x \in {\bf{R}} \cr} $$