Suppose a satisfies 4 = a + a^{-1}. What is the value of a^4 + a^{-4}?

[nanase]

Hellow, I need help in solving this expression :

I tried converting to quadratic or quartic equation, but can't proceed any further.

thank you for help

 

[topsquark]

Hellow, I need help in solving this expression :
View attachment 36683
I tried converting to quadratic or quartic equation, but can't proceed any further.
View attachment 36684
thank you for help

If [imath]a + \dfrac{1}{a} = 4[/imath], then what is a?

-Dan

 

[nanase]

If [imath]a + \dfrac{1}{a} = 4[/imath], then what is a?

-Dan

sorry? a is a variable right?

 

[nanase]

I (cheated) a bit by using desmos to graph and the intersection happens to be around 0.27 and 3.73.
I am not supposed to use technology and solve this manually? anybody can give hint?

 

[topsquark]

sorry? a is a variable right?

I gave you a big hint. And, yes, a is a variable.

A further hint: multiply both sides of the equation by a, and then you have a quadratic equation.

Look, not being good at Mathematics is no crime, but continually trying to do problems over your head is. May I ask: what level of Mathematics are you at? If you are trying problems like these, you should not need to ask if a is a variable!

-Dan

 

[khansaheb]

Hellow, I need help in solving this expression :
View attachment 36683
I tried converting to quadratic or quartic equation, but can't proceed any further.

thank you for help

Can you find the expression for (a + a^-1)² --- using (x + y)² = x² + y² + 2*x*y

then

Can you find the expression for (a² + a^-2)² --- using (x + y)² = x² + y² + 2*x*y .................. continue

 

[Steven G]

Do you know the value of a? If no, then a is a variable. You were on the right track by getting that quadratic equation so why did you not continue??

 

[Steven G]

4 = ( a+ 1/a)
256 = a^4 + 4a^2 + 6 + 4/a^2 + 1/a^4
256= (a^4 + 1/a^4) + 4a^2 + 6 + 4/a^2
250= (a^4 + 1/a^4) + 4a^2 + 4/a^2
16 = a^2 + 2 + 1/a^2
14 = a^2 + 1/a^2
Continue from here.

 

[Dr.Peterson]

Hellow, I need help in solving this expression :
View attachment 36683
I tried converting to quadratic or quartic equation, but can't proceed any further.
View attachment 36684
thank you for help

If you just square the equation twice, you will get the answer (with a little manipulation in the middle). See post #6.

 

[nanase]

I am getting something like this. I am stuck on how to proceed

 

[Steven G]

Do you know what a^2 + 2 + 1/a^2 equals?
Now how much does a^2 + 1/a^2 equals?
Lastly, how much does 4a^2 + 4/a^2 equal?

 

[nanase]

oh after doing some algebra ---> a^2 + 1/a^2 = 14
I multiply all by 4 which gives 56
so subtract that from 250
Thank you Mr Steven!

 

[Steven G]

oh after doing some algebra ---> a^2 + 1/a^2 = 16
I multiply all by 4 which gives 56
so subtract that from 256
Thank you Mr Steven!

First a^2 + 1/a^2 is not 16.

Just look at your first line in post 10!!!
It says that (a^2 + 2 + 1/a^2)^2 = 16^2 So what does a^2 + 2 + 1/a^2 equal?? The answer is right there!

 

[nanase]

Ah I made some careless typo & calculations, I edited my #12 hopefully that works
250-56 =194

 

[Steven G]

Ah I made some careless typo & calculations, I edited my #12 hopefully that works
250-56 =194

I'm not saying arriving at a^2 + 1/a^2 = 14 is wrong but I wonder after arriving at what you got on post 10, how did you get a^2 + 1/a^2 = 14??

 

[nanase]

I'm not saying arriving at a^2 + 1/a^2 = 14 is wrong but I wonder after arriving at what you got on post 10, how did you get a^2 + 1/a^2 = 14??

I followed up on Dr Peterson advice to do some manipulation in the middle, from squaring the equation into quadratic

 

[khansaheb]

a^2 + 1/a^2 = 14

Then

(a^2 + 1/a)^2 = (14}^2

a^4 + 2 * a^2 * (1/a)^2 + (1/a^4)= 196

a^4 + (1/a)^4 = 196 - 2.......................continue

 

[Steven G]

I followed up on Dr Peterson advice to do some manipulation in the middle, from squaring the equation into quadratic

You already had in your prior post that (a^2 + 2+ 1/a^2) = 16^2. That means that a^2 + 2+ 1/a^2 =1 6. From there a^2 +1/a^2 = 14 (14 + 2 = 16).