What's the probability that the roots of $4x^2 + 4xK + K + 2 = 0$ are real?

Question

If the random variable $K$ is uniformly distributed over $(0, 5)$, what is the probability that the roots of the equation $4x^2 + 4xK + K + 2 = 0$ are real. $Awnser: \frac{3}{5}$

This is the question 4.28 from (Paul Meyer's "Introductory probability and Statistics", 2nd ed.)

What I've managed to do was find the density function, which is: $\frac{1}{5}$. And since it's a quadratic equation, $\Delta$ must be greater or equal to zero to achieve real roots, in this case $\Delta = 4K^2 - 4(K + 2)$ and the question is asking for the $P(\Delta ≥ 0)$ therefore.

My trouble is i don't know how to compute this, or even if my assumptions are right.

Answer 1

Why you don't know how to compute that? $\Delta=(4K)^2-4\cdot4\cdot(K+2)=16(K-2)(K+1)$, so if $K\in(0,5)$, then $\Delta\ge0 \Leftrightarrow 2\le K<5$.

Probability density function of $K$ is $f(K)=\frac15$, so...

$$\Bbb P(\Delta\ge0)=\Bbb P(2\le K<5)=\int_2^5f(K)dK=\int_2^5\frac15dK=\frac35$$