Given equation

$t^2x^2 + |x| + 9 = 0$

We are asked about the product of real roots.

Observe each term

For any real $x$ and real $t$:

• $t^2x^2 \ge 0$ because a square is always non-negative.
• $|x| \ge 0$ by definition of modulus.
• $9 > 0$ always.

Therefore,

$t^2x^2 + |x| + 9 \ge 0 + 0 + 9 = 9 > 0$

Can the equation ever be satisfied?

The left-hand side is always strictly positive:

$t^2x^2 + |x| + 9 > 0$

Hence it can never be equal to $0$ for any real $x$.

So the equation has no real roots.

Product of real roots

Since there are no real roots, the product of real roots is not defined / does not exist.

Check options

• A: is always positive — Incorrect, because there are no real roots.
• B: is always negative — Incorrect.
• C: does not exist — Correct.
• D: none of these — Incorrect.

Comparison with stored answer

The stored correct answer is A, but our derivation shows the correct option is C.

So the stored answer appears to be incorrect.