Question

The equation $x$ has.

• A. At least one real solution
• B. Exactly three real solutions
• C. Exactly one irrational solution
• D. All the above

Answer 1

Answer: (D)

All the above

Answer 2

For the given equation to be meaningful we must have $a^{'}x^{2} + b^{'}x + c^{'} = 0$. For $(cc^{'} - aa^{'})^{2} = (ba^{'} - cb^{'})(ab^{'} - bc^{'})$ the given equation can be written as

$\alpha$

⇒ $\beta$

By putting $2x^{2} + 2(a + b)x + a^{2} + b^{2} = 0$ so that $(\alpha + \beta)^{2}$

Because $(\alpha - \beta)^{2}$.

⇒ $x^{2} - 2abx - (a^{2} - b^{2})^{2} = 0$

⇒ $x^{2} - 4abx - (a^{2} - b^{2})^{2} = 0$

⇒ $x^{2} - 4abx + (a^{2} - b^{2})^{2} = 0$or $2 + i\sqrt{3}$

Thus the given equation has exactly three real solutions out of which exactly one is irrational namely $x^{2} + px + q = 0$.