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Chapter 5: Q 5.38 (page 214)

If Yis uniformly distributed over (0,5),what is the probability that the roots of the equation 4x2+4xY+Y+2=0are both real?

Short Answer

Expert verified

Thus, the probability of real roots isP{Y鈮�2}=35

Step by step solution

01

Given information:

Yis uniformly distributed on(0,5)

02

Explanation:

The two roots of the quadratic are:

x=-416Y2-16(Y+2)8=12-1卤Y2-Y-2

The roots are real if and only ifY2-Y-2鈮�0.consider next the roots of the equationy2-y-2=0. Then solution isy=1卤1+82=1卤32=-1or2

03

EXPLANATION:

This means thaty2-y-2=(y+1)(y-2)which can be checked directly too. Consequently,Y2-Y-2鈮�0if and only ifY-2鈮�0. Thus, the probability of real roots isP{Y鈮�2}=35

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