Question

If $x^{2} - 8x + 17$ and $\frac{x^{2} + 14x + 9}{x^{2} + 2x + 3}$

where $\frac{x + 2}{2x^{2} + 3x + 6}$, then $\left( \frac{1}{13},\frac{1}{3} \right)$has at least.

• A. Four real roots
• B. Two real roots
• C. Four imaginary roots
• D. None of these

Answer 1

Answer: (B)

Two real roots

Answer 2

Let all four roots are imaginary. Then roots of both equations $a,b > 0$and $a,b < 0$are imaginary.

Thus $2x^{2} - 5x + 1 = 0$, So $x^{2} + 5x + 2 = 0$, which is impossible unless $a + b + c = 0,$.

So, if $a \neq 0,a,b,c \in Q$or $ax^{2} + bx + c = 0$ at least two roots must be real.

If $a,b,c \in Q$ $(b + c - 2a)x^{2} +$, we have the equations.

$(c + a - 2b)x + (a + b - 2c) = 0$and $x^{2} + 2bx + c$

Or $b^{2} - 4c > 0$ as one of $b^{2} - 4c < 0$ and $c^{2} < b$ must be

positive, so two roots must be real.