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Let a,b,c   be positive real numbers. Determine the largest total number of real roots that the following three polynomials may have among them:   ax^2+bx+c,  bx^2+cx+a,  cx^2+ax+b

 

Please help!!!

 Nov 26, 2019
 #1
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Each quadratic can have two real roots, so the maximum number is 6 real roots.

 Nov 27, 2019
 #2
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i answered that but it was wrong.... help! Any other answers I can achieve? it wasn't 6 

 Nov 27, 2019
 #3
avatar+2490 
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Each of these quadratics have a maximum of 2 solutions as long as their discriminant works.

 

try creating a system of inequalities using the properties of a discriminant with a, b, c??????

 

 

Not sure if that helped I don't have much time to solve this because I have a math class in literally 5 min.

 Nov 27, 2019
 #4
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thanks! the trick actually was that a,b,c were positive. the answer was 4

 Nov 27, 2019
 #5
avatar+105971 
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Mmm

 

ax^2+bx+c,  bx^2+cx+a,  cx^2+ax+b

 

\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\\ x = {-c \pm \sqrt{c^2-4ab} \over 2b}\\ x = {-a \pm \sqrt{a^2-4cb} \over 2c}\\ \)

 

These are the zeros for the 3 expressions.  Off hand I cannot see why there cannot be 6 roots in total.......

Ok I graphed it in desmos and if does not appear to be able to have 6 roots.

I can get 4 but I could work out how to get more.

https://www.desmos.com/calculator/hrlzvwageo

 

 

 

For any of these to have two roots, the desciminant must be greater than 0.

That can certainly be true for at least one of these but can it be true of 2 or for all there.

Obviously it can be true for 2 cause I graphed it.

 

So I really do not have a definitive answer.    Most likely 4 but I have not seen any maths to prove it.

 Nov 27, 2019

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