Numbers $\alpha$ and $\beta$ are selected from interval $[0,1]$. What is the probability that $x^2+\alpha x + \beta ^2=0$ has real roots?

Question

I know that discriminant must be greater than zero , so we have :

$\alpha ^2-4\beta^2\geq 0$

$\alpha^2\geq4\beta^2$

$\alpha\geq 2\beta$

We draw a function $\alpha - 2\beta = 0 $ and our condition is there where that function is greater than zero beetween the segment and we get the conditional area.

I don't know have to calculate the total area $\Omega$ which I need to get the probability.

Some version of this question has been asked here many times, for instance: http://math.stackexchange.com/questions/29242/probability-that-a-quadratic-polynomial-with-random-coefficients-has-real-roots — colormegone, Sep 02 '15 at 19:02

score 2 · Accepted Answer · answered Sep 02 '15 at 18:36

2

If we draw the square $[0,1]\times[0,1],$ we can see that the segment $\alpha=2\beta$ cuts the square into a trapezoid and a triangle. Which of those is the relevant region? What is its area? Do you see how this is related to the answer?

answered Sep 02 '15 at 18:36

Cameron Buie

102,994

Thanks :) the squre was a problem, I thought it is only a segment on the x - axis. Silly me :) – Sep 02 '15 at 19:07

Numbers $\alpha$ and $\beta$ are selected from interval $[0,1]$. What is the probability that $x^2+\alpha x + \beta ^2=0$ has real roots?

1 Answers1