Comparing Solutions of Quadratic Equations: Real vs Imaginary Roots

AI Thread Summary
The discussion focuses on solving quadratic equations with real and imaginary roots. The initial calculations show a method of isolating x and taking the square root, resulting in imaginary solutions. A comparison with the quadratic formula reveals a different approach, leading to a discriminant of -140 and further simplification. The participant acknowledges a Latex error and realizes a mistake in their calculations, specifically not following through the steps completely. The conversation concludes with gratitude for the assistance received.
hackedagainanda
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Homework Statement
Solve for x, 7x^2 + 5 = 0
Relevant Equations
##x = \frac {-b \pm \sqrt{b^2 -4ac}} {2a},##
I subtract 5 from both sides to get 7x^2 = -5 Then I divide both sides by 7 to get -5/7. I then take the square root to get x = sqrt of the imaginary unit i 5/7 then ##\pm { i \sqrt \frac 5 7}##

The quadratic formula on the other hand gets me a different answer, the discriminant = -140 which can be simplified to 2 sqrt 35 i over 14 and then you factor out the 2 and get ##\pm i \sqrt \frac {35} 7##I see there is a Latex error but the root is only in the numerator
Both answers seem correct to me I don't see my error.
 
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b=0, so:
$$x = \pm \frac{\sqrt{-4*35}}{2*7}$$
$$x = \pm \frac{2\sqrt{-35}}{2*7}$$
$$x = \pm \sqrt{\frac{-35}{49}} = \pm \sqrt{\frac{-5}{7}}$$
 
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That's the answer I got, I take it the book rationalized the denominator. I see my error now, I didn't follow the steps all the way through.
 
Thanks for the help! You are very appreciated :smile:
 
I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
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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