It is not a difficult equation. I wonder, how people approach it. Here it is:

\[x+\frac{1}{x}=4\frac{1}{4}\]

(Please, use spoiler in your response.)

Spoiler

Method 1:

4 and 1/4 can be spotted directly, as 4+1/4 = 1/4+1/(1/4), and it's clearly equivalent to a quadratic equation (once zero is ruled out as a solution), so those are the only roots.

Method 2:

I rule out x=0, as it is not a solution. Then I mulltiply by x to obtain the quadratic eq. 4x²-17x+4=0, which gives x=4 and x=1/4 via the algorithm for solving quadratic equations.

Spotting obvious solutions first can sometimes be very helpful to obtain the non-obvious ones when the equation is polynomial and has degree > 4.

Example: x⁵=x

What would you do?

Spoiler

First, x=0. Then, divide by x to get x⁴=1 and thus x=1, -1, i, -i.

 

Edited November 3, 20241 yr by Genady

Spoiler

x=4 is one obvious root just on inspection, but if we ignore that:

Multiply both sides by 4x to give 4x² + 4 = 17x or 4x² - 17x+ 4 = 0

By formula x =(17 +/- (17² - 4³)^1/2)/8

Ans x = 4 or 1/4

Ooh! That's neat!

... what else could you be looking for?

11 minutes ago, sethoflagos said:

  Reveal hidden contents

x=4 is one obvious root just on inspection, but if we ignore that:

Multiply both sides by 4x to give 4x² + 4 = 17x or 4x² - 17x+ 4 = 0

By formula x =(17 +/- (17² - 4³)^1/2)/8

Ans x = 4 or 1/4

Ooh! That's neat!

... what else could you be looking for?

Spoiler

Many go straight to 17/4 and miss the first approach. OTOH, some see both answers by the first approach.

 

Edited November 3, 20241 yr by Genady

Spoiler

For me, one solution popped out by seeing that 4 1/4 can be said as "4 and one fourth."  So then x + 1/x is obviously 4. 

Then say the right hand part backwards, "one fourth and four" and you get the other solution, 1/4.

It's really just noticing the RH part is just a number and its reciprocal.  

 

31 minutes ago, sethoflagos said:

what else could you be looking for?

You did it the hard way.  🙂

4 minutes ago, TheVat said:

  Hide contents

 

You did it the hard way.  🙂

Force of habit. I ALWAYS do the check longhand just to be sure. (Chem Eng thing)

5 hours ago, sethoflagos said:

  Reveal hidden contents

x=4 is one obvious root just on inspection, but if we ignore that:

Multiply both sides by 4x to give 4x² + 4 = 17x or 4x² - 17x+ 4 = 0

By formula x =(17 +/- (17² - 4³)^1/2)/8

Ans x = 4 or 1/4

Ooh! That's neat!

... what else could you be looking for?

I just thought of something else, for a case when one sees one solution but not the other:

Spoiler

Instead of solving the quadratic equation, one can notice that after multiplying by x the quadratic equation will have a constant term 1, which means that the product of the roots is 1, which gives the second root, 1/4.

 

2 hours ago, Genady said:

I just thought of something else, for a case when one sees one solution but not the other:

  Reveal hidden contents

Instead of solving the quadratic equation, one can notice that after multiplying by x the quadratic equation will have a constant term 1, which means that the product of the roots is 1, which gives the second root, 1/4.

 

My impression of equations of that type are dominated by:

x - 1/x = 1

12 minutes ago, sethoflagos said:

My impression of equations of that type are dominated by:

x - 1/x = 1

Could you elabotate?

9 hours ago, Genady said:

Could you elabotate?

Spoiler

Your OP equation is one of a family whose roots are the positive inverse of each other.

Conversely mine is one of a family whose roots have a negative inverse relationship. 

eg. x - 1/x = 3 ³/₄ solves to x = 4, -¹/₄

x - 1/x = 1 solves to the golden ratio (as I'm sure you know) and its negative inverse. A deep, deep rabbit hole I've ventured into on occasion.

 

Compare and contrast:

x + ¹/_x = 5^1/2

Edited November 4, 20241 yr by sethoflagos

1 hour ago, sethoflagos said:

  Reveal hidden contents

Your OP equation is one of a family whose roots are the positive inverse of each other.

Conversely mine is one of a family whose roots have a negative inverse relationship. 

eg. x - 1/x = 3 ³/₄ solves to x = 4, -¹/₄

x - 1/x = 1 solves to the golden ratio (as I'm sure you know) and its negative inverse. A deep, deep rabbit hole I've ventured into on occasion.

 

Compare and contrast:

x + ¹/_x = 5^1/2

I see.

A general form

Spoiler

\[x+\frac{a}{x}=b\]

has roots with the product \(a\) and the sum \(b\). 

 

(However, the OP was about how people approach that equation rather than a mathematics behind it.)

(I see how YOU approach it 🙂 )

Edited November 4, 20241 yr by Genady

I'll post my approach, for the record.

Spoiler

I wrote the RHS as \[x+\frac{1}{x}=4+\frac{1}{4}\]

This immediately gave me one answer.

The second answer took a bit more effort, but I don't recall the exact thinking there.