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1) What is the smallest distance between the origin and a point on the graph of
[latex added, bobbym]
2) Determine the sum of all real numbers x satisfying (x^2-4x+2)^{x^2-5x+2}=1.
3)Let a, b be real numbers, and let x_1, x_2 be the roots of the quadratic equation x^2+ax+b=0. Prove that if x_1, x_2 are real and nonzero, \frac 1{x_1}+\frac 1{x_2}<1, and b>0, then |a+2|>2.
Only have a lead on #2:
I let y=x^2-4x+2 so that the equation would read y^{y-x}=1. Since y^0=1 and they have the same base, I knew that y=x. Substituting again, I would get that x^2-4x+2=x. Subtracting x from both sides, I then used the quadratic formula to find the roots of x. But once I added the roots together, the answer was wrong. Can anyone tell me what I'm doing wrong? (I got 5.)
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Hi;
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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For
, we have to consider the cases when . Most notably, we have b=0, a=1 as the obvious ones.For the third problem,
which you can use Vieta's formula...Offline
Last edited by thickhead (2016-11-07 19:47:55)
{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}
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Hi;
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}
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I think in Q.3 there is something missing. What exactly is ths? \frac 1{x_1}+\frac 1{x_2}<1
evene's interpretation does not seem to be what is intended or something is missing.
{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}
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Thanks for the other hints! But I still can't figure out 2.
\frac 1{x_1}+\frac 1{x_2}<1 is
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Hi;
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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I think the only roots are
, , , andI still don't understand the 3rd question with Vieta's formula either
Last edited by dazzle1230 (2016-11-09 10:08:59)
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Hi;
I think the only roots are...
The question calls for you to add up those roots.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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The sum would be 9.
9 isn't right though
I still don't know how to proceed the 3rd question though
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I think the only roots are
, , , and
Then that statement is incorrect.
You are not considering all the possibilities or at least enough.
(x^2-4x+2)^{x^2-5x+2}=1.
1) You could have 1^{x^2-5x+2}=1 that would mean (x^2-4x+2) = 1
2) You could have (x^2-4x+2)^0 =1 that would mean (x^2-5x+2) = 0
3) You could have (-1)^(2n) = 1 that would mean (x^2-4x+2) = -1 and (x^2-5x+2) is even.
So solve (x^2-4x+2) = -1 and add the new roots to the ones already found and see what happens then.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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Thank you! Its right now.
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Hi;
Yep, the answer is the ugly 13. But it could have been much uglier had you not been able to check against what you know the right answer is.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
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I need help on 3:
I know that a>-1, the product of the roots is greater than 1, and the discriminant is greater than 0. That's about it
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I proved that a>0, but I'm having trouble proving that a<-4
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{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}
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It was actually a>0, sorry
Is that for proving a<-4?
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discriminant a^2-4b>0 but 4a>-4b So a^2+4a>0 i.e. a^2+4a+4>4; (a+2)^2>4 which means |a+2|>2
I think my hint did not provoke you.
{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}
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