Question
For the equation $$\left| {{x^2}} \right| + \left| x \right| - 6 = 0,$$ the roots are
A.
One and only one real number
B.
Real with sum one
C.
Real with sum zero
D.
Real with product zero
Answer :
Real with sum zero
Solution :
When $$x < 0,\left| x \right| = - x$$
$$\therefore $$ Equation is $$x^2 - x - 6 = 0$$
⇒ $$x = - 2, 3$$
$$\because $$ $$x < 0,$$ $$\therefore $$ $$x = - 2$$ is the solution.
When $$x \geqslant 0,\left| x \right| = x$$
$$\therefore $$ Equation is $$x^2 + x - 6 = 0$$
⇒ $$x = 2, - 3$$
$$\because x \geqslant 0,\therefore x = 2$$ is the solution,
Hence, $$x = 2, - 2$$ are the solutions and their sum is zero.