- #1

swampwiz

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This seems to just be the quadratic formula in a transposed way.

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- #1

swampwiz

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This seems to just be the quadratic formula in a transposed way.

- #2

jedishrfu

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The is not unlike a similar symmetry found in complex roots of a polynomial being equally spaced about a circle in the complex plane aka DeMoivre’s theorem.

https://en.m.wikipedia.org/wiki/De_Moivre's_formula

i remember using the symmetry in a test because i couldn't remember the formula but did have one real root of the 5th order polynomial and we were to graph the solution.

- #3

FactChecker

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- #4

jedishrfu

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When I learned to factor they didnt teach that instead you were looking for two numbers which summed correctly and when multiplied got the c term. It seems that root summing symmetry gets you to an answer more quickly.

- #5

Mark44

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The technique described is not really all that different from what happens in the technique of completing the square. Let's look at an example, such as solving the equation ##x^2 - 4x - 5 = 0##The difference I see though is going to the sum and saying the roots are equidistant from half the sum.

If we approach this using completing the square, we have

##x^2 - 4x = 5##

##x^2 - 4x + 4 = 5 + 4##

##(x - 2)^2 = 9##

So either x - 2 = 3 or x - 2 = -3, producing the solutions x = 5 or x = -1.

The only difference I see is the substitution of ##u## for ##x - 2##, so ##u^2 = 9 \Rightarrow u = \pm 3##.

Undo the substitution, and you get the same solutions I showed.

- #6

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It seems like a geometric interpretation of the algebraic process of completing the square. I find the algebraic process to be direct enough that a geometric interpretation is only distracting. But people who are not as comfortable with the algebraic process might benefit from a geometric interpretation.The technique described is not really all that different from what happens in the technique of completing the square.

- #7

uart

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I wouldn't want to fully replace teaching completing the square though, as that's also very useful for things other than solving quadratics. For example putting an integral like this one into a more amenable form.

[tex] \int \frac{1}{x^2 - 2x + 10} \, dx = \int \frac{1}{(x-1)^2 + 3^2} \,dx[/tex]

BTW. I've used this technique many times in the past. Usually in the context of students first learning to factorize simple integer coeff quadratics in the form [itex]x^2 + bx + c[/itex] with the old "sum = b and product = c" method.

After giving the usual simple exercises like [itex]x^2 + 8x + 15\,[/itex] or [itex]x^2 + 5x + 6[/itex], better students will sometimes ask, "what if you just cannot find any two such numbers with the required sum and product?". The answer I give is that either no such numbers exist (no real roots) or that they exist but are surds and hence more difficult to find. I give an example like [itex]x^2 + 6x +7\,[/itex] and tell them to try (-3+s) and (-3-s), where "s" is the surd to be determined. I never realized I'd found a "new process".

- #8

symbolipoint

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I found the simple geometric method to be extremely helpful, to ME at least, if not to other people.I find the algebraic process to be direct enough that a geometric interpretation is only distracting. But people who are not as comfortable with the algebraic process might benefit from a geometric interpretation.

- #9

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I'll buy that.I found the simple geometric method to be extremely helpful, to ME at least, if not to other people.

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