Solving Quadratic Eqn for Symbolic Complex Eqn

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The discussion revolves around solving a complex quadratic equation with specific coefficients. The user is attempting to analytically simplify the quadratic equation using the quadratic formula but is struggling with the discriminant. They aim to factor the expression under the square root into a recognizable form, ultimately seeking to express it as a perfect square. Another participant suggests that the polynomial can be rearranged into a specific squared form, which helps the original poster realize their mistake. The user successfully identifies the correct form after considerable effort, highlighting the complexity of the problem.
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Homework Statement



Hey guys, I know there are people on here that can help me figure this out...

I have a complex quadratic. The coefficients of the quadratic are a=(1-x) b=x*(u+P/(u*r)) and c=-(u^2 + x*P/r). Using -b/2a +/- sqrt(b^2-4*a*c)/2a

I'm trying to analytically reduce this quad eqn as far as I can

I know this should reduce to 1/(x-1)*u + x*P/(r*u)

The Attempt at a Solution



As far as I could take it was trying to reduce the term in the sqrt => b^2-4*a*c

u^2*(x-2)^2 + (P/(u*r))^2*(x)^2 + P/r*(-4x^2 + 4x)

I feel like I should be able to factor this into something like

(u+p/(u*r))^2 * (something)^2 so I can take the sqrt but I can't figure this out. Thanks in advance for the help.

~ Wesley
 
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WesleyJA81 said:

The Attempt at a Solution



As far as I could take it was trying to reduce the term in the sqrt => b^2-4*a*c

u^2*(x-2)^2 + (P/(u*r))^2*(x)^2 + P/r*(-4x^2 + 4x)
Don't write this. Leave it in the "unfactored" form for a moment. There should be 6 terms, three of which are perfect squares. It will be in the form of
a^2 + b^2 + c^2 + 2ab + 2bc + 2ac = (a + b + c)^2

See if you can get the polynomial in the above form.


(Mods: I hope that this isn't too much of help. If so, please delete.)
 
eumyang, thanks for the tip. Let me take a look. The problem I'm working on is bigger than what I've posted. This is just the point that has stumped me. Thanks again!
 
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eumyang said:
Don't write this. Leave it in the "unfactored" form for a moment. There should be 6 terms, three of which are perfect squares. It will be in the form of
a^2 + b^2 + c^2 + 2ab + 2bc + 2ac = (a + b + c)^2

See if you can get the polynomial in the above form.


(Mods: I hope that this isn't too much of help. If so, please delete.)


It ended up being of the form

a^2 + b^2 + c^2 - 2ab + 2bc - 2ac = (-a + b + c)^2

I'm embarrassed at the amount of time I spent trying to figure that out. Thanks again
 
Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks
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