What is the probability that the quadratic equation a x 2 + x + 1 = 0 has two real roots?A number a is chosen at random within the interval (-1,1). What is the probability that the quadratic equation a x 2 + x + 1 = 0 has two real roots?For it to have its real roots, we must guarantee that 1 − 4 a ≥ 0, or a ≤ 1 4 .

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What is the probability that the quadratic equation   a      x    2    +  x  +  1  =  0 has two real roots?A number a is chosen at random within the interval (-1,1). What is the probability that the quadratic equation   a      x    2    +  x  +  1  =  0 has two real roots?For it to have its real roots, we must guarantee that   1  −  4  a  ≥  0, or   a  ≤      1    4  .

dilettato5t1 · Accepted Answer

Step 1We want the probability that   a  ∈  (  −  1  ,      1    4    ] given that it is uniformly chosen from the interval (-1,1).Step 2Since the interval   (  −  1  ,      1    4    ) has length       5    4   and the interval (-1,1) has length 2, the probability is             5      4        2  Answer:                                           5            8

Ricky Arias · Accepted Answer

Step 1This equations have two distinct real solutions iff   a  ∈  (  −  1  ,  0  )  ∪  (  0  ,      1    4    ). (When   a  ∈  {  0  ,      1    4    } it has one real solution) Therefore the probability is   P  =            l      (      (      −      1      ,      0      )      ∪      (      0      ,              1        4            )      )              l      (      −      1      ,      1      )      Step 2Here l(I) is the length of interval I.  P  =            1      +              1        4              2    =      5    8

What is the probability that the quadratic equation ax^2+x+1=0 has two real roots?

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