Question

Product of real roots of the equation ${{t}^{2}}{{x}^{2}}+|x|+9=0$
A . is always positive
B. is always negative
C. does not exists
D. none of these

Answer 1

Hint: In this question, we are given a quadratic equation and we have to find out the product of the roots of the equation satisfies which option. We solve this question logically. A square of a negative or a positive number always gives us a positive value. So the addition of all the values gives us a positive value. On behalf of that, we find out the option that satisfies.

Complete step by step Solution:
Given quadratic equation is ${{t}^{2}}{{x}^{2}}+|x|+9=0$
We have to that the product of their roots satisfies which option.
Let us assume that t and x are real
From the equation ${{t}^{2}}{{x}^{2}}+|x|+9=0$
${{t}^{2}}{{x}^{2}}\ge 0$
As we square a positive number, then it always is positive and if we square a negative number, then it also is positive.
So ${{t}^{2}}{{x}^{2}}\ge 0$
Also $|x|>0$
We know the mod function also gives us a positive value.
So $|x|>0$
And we know 9 is also a positive number.
So $|9|>0$
So on adding all these equations, we get
${{t}^{2}}{{x}^{2}}+|x|+9>0$
But in question adding positive values gives us zero.
As long as t and x are assumed real, hence there exists no real root.
Therefore the product of the equation ${{t}^{2}}{{x}^{2}}+|x|+9=0$ does not exist.

Therefore, the correct option is (C).

Note:We know in a modulus function
$f(x) = x if x ≥ 0 or -x if ≤ 0. $
Adding all the positive values gives us a positive value and that positive value is> 0. And the question tells us all the values = 0. So no real root exists.