Question

If $Y$is uniformly distributed over $(0,5),$what is the probability that the roots of the equation $4x^2+4xY+Y+2=0$are both real?

Short answer

Thus, the probability of real roots is$P\{Y\ge 2\}=\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$5$}\right.$

Step-by-step solution

Given information:

$Y$is uniformly distributed on$(0,5)$

Explanation:

The two roots of the quadratic are:

$x=\frac{-4\sqrt{16Y^2-16(Y+2)}}{8}=\frac{1}{2}\left[-1\pm \sqrt{Y^2-Y-2}\right]$

The roots are real if and only if$Y^2-Y-2\ge 0.$consider next the roots of the equation$y^2-y-2=0$. Then solution is$y=\frac{1\pm \sqrt{1+8}}{2}=\frac{1\pm 3}{2}=-1or2$

EXPLANATION:

This means that$y^2-y-2=(y+1)(y-2)$which can be checked directly too. Consequently,$Y^2-Y-2\ge 0$if and only if$Y-2\ge 0$. Thus, the probability of real roots is$P\{Y\ge 2\}=\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$5$}\right.$