How do YOU solve this equation?

[Genady]

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.)

[joigus]

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?

[Genady]

Spoiler

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

 

Edited November 3, 2024 by Genady

[sethoflagos]

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?

[Genady]

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, 2024 by Genady

[TheVat]

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.  🙂

[sethoflagos]

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)

[Genady]

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.

 

[sethoflagos]

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

[Genady]

12 minutes ago, sethoflagos said:

My impression of equations of that type are dominated by:

x - 1/x = 1

Could you elabotate?

[sethoflagos]

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, 2024 by sethoflagos

[Genady]

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, 2024 by Genady

[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.