Let a , b , c be real. If ax^2 + bx + c = 0 has two real roots α and β where …..

Q: Let a , b , c be real. If ax^2 + bx + c = 0 has two real roots α and β where α < -1 and β > 1, then show that $\displaystyle 1 + \frac{c}{a} + |\frac{b}{a}| < 0 $

Sol: Let $\displaystyle f(x) = x^2 + \frac{b}{a} x + \frac{c}{a} $

On drawing graph f(-1) < 0 and f(1) < 0

$\displaystyle 1 + \frac{c}{a} – \frac{b}{a} < 0 $

and , $\displaystyle 1 + \frac{c}{a} + \frac{b}{a} < 0 $

⇒ $\displaystyle 1 + \frac{c}{a} + |\frac{b}{a}| < 0 $