Show that the roots of the equation $x^{2} + p x - q^{2} = 0$ are real for all real values of p and q

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Solution

Given equation is:

x² + px – q² = 0

The discriminant of the given equation is given by

D = p² – 4 × (1) × (–q²)

= p² + 4q²

Clearly, D = p² + 4q² > 0 for all p, q ∈ R.

Hence, the given equation has real roots.

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