Solve the given quadratic equation that involves sum and product

AI Thread Summary
The discussion focuses on solving a quadratic equation using the relationships between the sum and product of its roots, denoted as α and β. Part a derives a formula involving the roots, leading to the expression (c-1)² + b². In part b, a quadratic equation is formulated with coefficients derived from the roots and their relationships. Participants confirm the correctness of the approach and seek alternative methods for solving the problem. Overall, the thread emphasizes the mathematical derivation and verification of the quadratic equation based on the given parameters.
chwala
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Homework Statement
See attached below;
Relevant Equations
sum/product
1644621814964.png


For part a,
We have ##α+β=b## and ##αβ =c##. It follows that,
##(α^2 + 1)(β^2+1)=α^2β^2+α^2+β^2+1)##
=##α^2β^2+(α+β)^2-2αβ +1##
=##c^2+b^2-2c+1##
=##c^2-2c+1+b^2##
=##(c-1)^2+b^2##

For part b,..we shall have
##x^2- \dfrac{α+β+α^2β +αβ^2}{(α^2 + 1)(β^2+1)}## ##x## +##\dfrac {αβ}{(α^2 + 1)(β^2+1)}##
##x^2-\dfrac{α+β+αβ(β +α)}{(α^2 + 1)(β^2+1)}####x##+##\dfrac {αβ}{(α^2 + 1)(β^2+1)}##
##x^2-\dfrac{b(1+c)}{(c-1)^2+b^2}####x##+##\dfrac {c}{(c-1)^2+b^2}##

Is this correct?( i do not have the solutions)...i would appreciate different ways of attempting the problem. Cheers.
 
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It seems all right.
 
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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