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#1 2018-03-13 11:38:34

Mathegocart
Member
Registered: 2012-04-29
Posts: 2,226

Somewhat confused on a quadratic question.

Hi, amiable friends, and I took a quadratics test today. I was somewhat confused by this question.
If the coefficients of a quadratic equation are real, and if one of the roots is (1-sqrt(3)), the other root must be (1+sqrt(3)).
TRUE?
FALSE?
This shouldn't be true, should it? The other root can be anything, and the coefficents would still be real.

Last edited by Mathegocart (2018-03-13 11:39:06)


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#2 2018-03-13 17:59:51

zetafunc
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Registered: 2014-05-21
Posts: 2,436
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Re: Somewhat confused on a quadratic question.

False: consider the equation
.

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#3 2018-03-15 16:22:07

Jai Ganesh
Administrator
Registered: 2005-06-28
Posts: 48,429

Re: Somewhat confused on a quadratic question.

Hi,

False.

Other than

, there are any number of possibilities. Some are
.

is one of them.


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#4 2018-03-15 21:58:40

Libera
Member
Registered: 2018-03-07
Posts: 16

Re: Somewhat confused on a quadratic question.

ganesh wrote:

Hi,

False.

Other than

, there are any number of possibilities. Some are
.

is one of them.

I'd find interesting to share the different ways we use to approach the problem. My immediate view was graphical, for instance. And yours algebraic, I suppose.
I "saw" a parabola "free to move" but its fixed point

Last edited by Libera (2018-03-15 23:43:23)

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#5 2018-03-15 23:06:13

Alg Num Theory
Member
Registered: 2017-11-24
Posts: 693
Website

Re: Somewhat confused on a quadratic question.

Mathegocart wrote:

If the coefficients of a quadratic equation are real, and if one of the roots is (1-sqrt(3)), the other root must be (1+sqrt(3)).
TRUE?
FALSE?

If the coefficients were rational, then the statement would be true. It’s false if you allow the coefficients to be irrational, as the replies below your post show.

If the coefficients are real and one of the roots is complex (and non-real), then the other root must be the complex conjugate. This statement also fails if you allow the coefficients to be complex.


Me, or the ugly man, whatever (3,3,6)

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#6 2018-03-16 00:24:14

Alg Num Theory
Member
Registered: 2017-11-24
Posts: 693
Website

Re: Somewhat confused on a quadratic question.

Alg Num Theory wrote:

If the coefficients were rational, then the statement would be true.


In general, if the coefficients are rational and one root is an irrational surd, then the other root must be the conjugate surd. This is not true if the coefficients can be irrational as well as rational.


Me, or the ugly man, whatever (3,3,6)

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