Math Is Fun Forum

dazzle1230

Member

Registered: 2016-05-17

Posts: 92

Algebra Problems

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.)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Algebra Problems

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.

evene

Member

Registered: 2015-10-18

Posts: 272

Re: Algebra Problems

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...

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Algebra Problems

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.}

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Algebra Problems

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.

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Algebra Problems

---

{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.}

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Algebra Problems

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.}

dazzle1230

Member

Registered: 2016-05-17

Posts: 92

Re: Algebra Problems

Thanks for the other hints!  But I still can't figure out 2.

\frac 1{x_1}+\frac 1{x_2}<1  is

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Algebra Problems

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.

dazzle1230

Member

Registered: 2016-05-17

Posts: 92

Re: Algebra Problems

I think the only roots are

,

,

, and

I still don't understand the 3rd question with Vieta's formula either

Last edited by dazzle1230 (2016-11-09 10:08:59)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Algebra Problems

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.

dazzle1230

Member

Registered: 2016-05-17

Posts: 92

Re: Algebra Problems

The sum would be 9.

9 isn't right though

I still don't know how to proceed the 3rd question though

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Algebra Problems

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.

dazzle1230

Member

Registered: 2016-05-17

Posts: 92

Re: Algebra Problems

Thank you! Its right now.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Algebra Problems

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.

dazzle1230

Member

Registered: 2016-05-17

Posts: 92

Re: Algebra Problems

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

dazzle1230

Member

Registered: 2016-05-17

Posts: 92

Re: Algebra Problems

I proved that a>0, but I'm having trouble proving that a<-4

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Algebra Problems

---

{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.}

dazzle1230

Member

Registered: 2016-05-17

Posts: 92

Re: Algebra Problems

It was actually a>0, sorry

Is that for proving a<-4?

thickhead

Member

Registered: 2016-04-16

Posts: 1,086

Re: Algebra Problems

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.}