I choose a random integer $n$ between $1$ and $10$ inclusive. What is the probability that for the $n$ I chose, there exist no real solutions to the equation $x(x+5) = -n$? Express your answer as a common fraction.
Hint, you shouldn't be lazy and let other people do the work for you, mathematicions like Albert Einstein took years to figure out one question.
anyways, you should simplify x(x+5)=-n which will be x^2+5x+n=0, then make a chart with two columns and 10 rows and you plug in 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 into x^2+5x+n=0, then if the discriminant (b^2-4ac) is negative, it has unreal roots,
I'll do one for you, for example if we have 7 as n, then
then plug these into b^2-4ac and you will get (5)^2 -- (4)(1)(7), this will equal to
25-28 which is [-3] this means that if n is 7, then the quadratic has no real roots, if b^2-4ac is equal to something positive then it has real roots.