okay let's do some math for fun and here i have a very interesting question for you guys given that
the equation square root of x plus square root of 20 minus x is equal to square root 20 plus 20 x
minus x squared so this radical equation has three square roots but anyway though
it has a total of four distinct real solutions and we are going to call them x1 x2 x3 x4
and um we can just assume that the x1 is the smallest and x2 is the next smallest and so
on and our goal is just to compute the sum this question is from the berkeley mini map
tournament in the year of 2021 and you know the deal please pause the video and try this first
done cool i really like this question because we do not have to find out all the solutions and
add them up you could but of course that would be longer in a math competition type of situation you
definitely want to think about the best way the nicest way the quickest way to solve the question
so let me show you guys how we can possibly do this check this out
inside here we have x but this right here is 20 minus x here we have 20 plus 20 x minus x squared
notice right here if we factor on x
then it seems like the inputs are kind of similar huh so let's go ahead and make an observation
so obs i'm going to keep the first square root of x the same likewise the next one
but for the third one let's factor the x so it's square root of 20 plus x times 20 minus x so from
here check this out i'm just going to put 20 minus x to be a another variable i'm just going to use
the same notation as the use in the solution then y be equal to 20 minus x and of course by doing
so you will also have to write the x in terms of y so we know we can put this there put that there so
put this there put this there so that means x is equal to 20 minus y and now here is
where the magic happens for the first part we can write this as square root and for the x
of course we know that's 20 minus y now and then next we have square root of 20 minus x which is
precisely our y and then continue we will get square roots and then this is 20 plus
the x is that so i will just put this as 20 minus y and lastly this right here is just y
this is so nice why because let's look at the structure of this equation
you see this input is square root of y yeah this is 20 minus the variable yeah in the
square root and we have 20 and then the x or just use y here by like 120 minus that variable
so what this means is that if x1 let's just use this right here for example is the solution
right then we also know that 20 minus x1 is also a solution
a solution
yeah i will give you guys a real quick example for example zero is a solution because if x is equal
to zero this is gong this is gone this is gone square root of 20 is equal to 20. but based on
what we said that means 20 minus 0 which is just 20. it's also a solution because if we put 20 here
put 20 here that will be gone that will be done square root 20 sql square 20.
so these are the two solutions right but of course the other two are
the harder ones binding way though right now what we can do is just say x1 plus x2 plus x3 and x4
for the x3 and x4 i'm just going to use that and again um it really depends how you want
to like write it out more properly but the question is just asking us to find the sum
so i'm just going to say this is the smallest and this is the next smallest yeah etc etc but then
for this one i'm just going to say is 20 minus x2 all right and then for this one i'm just going
to write it as 20 minus x1 because in fact this will be zero that's the smallest and this is the
biggest and yeah that's the same relationship in the middle anyway though you see how this
right here is so nice because this that can so this and that can so so the final answer just
20 plus 20. done yeah hey i hope you liked the video so far and if you enjoy problem solving and
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