Three numbers A,B,C are selected at random and independently fromthe interval (0

Question

Three numbers A,B,C are selected at random and independently fromthe interval (0,1). Determine the probability that thequadratic equation Ax2 + Bx + C has real roots. In otherwords what fraction of "all possible quadratic equations" withcoefficients in (0,1) have real roots?

Explanation / Answer

well first, we need to find out the requirements for real roots looking at the quadratic equation, for it to have real roots, theequations b2-4ac must be greater than or equal to 0, sob2>=4ac, since a, b, and c all have equal probabilityof having every possible number, we can say a=c, giving us b2>4a2, now we can take the square root ofboth sides, b>2a, now in this case, what are the chances of bbeing at least twice as big as a? lets split this up into 2cases, case 1, a>=0.5. In this case, b cannot be twice as big as a,since that would put it outside of the range. so that means that100% of the time, it won't have real roots for this case. case 2, a

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