Master SAT Math 💪| Find the value of phi+1/phi | Golden Ratio

In this video, we will find the value of phi+1/phi which golden ratio problem. This golden ratio problem is an SAT math
we will solve this in step-by-step. This is actually one of the challenging math problems so do not ignore this video.
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[EisFunnyLetter]

phi is when 1/phi=phi-1, and when you add that it equals the sqrt of 5. this is something we kinda memorise

[terribletruth]

Ay bro I'm from Bangladesh. And That was a beautiful math question...!

[aidenlim]

Once you get that phi + 1/phi = (phi² + 1)/phi, you can use the fact that
phi² - phi - 1 = 0
phi² + 1 = phi + 2
Then plug that in to the numerator to avoid squaring

[echandler]

phi is a root of the equation x^2 - x - 1 = 0. x is not zero. divide by x giving x-1/x = 1. Thus phi - 1/phi = 1. Consider the identity (a+b)^2 - (a-b)^2 = 4ab. Thus (a+b)^2 = (a-b)^2 + 4ab. Hence (phi+1/phi)^2 = (phi - 1/phi)^2 + 4 = 5. We know phi >0, thus phi + 1/phi > 0. Hence taking the positive square root, phi + 1/phi = sqrt(5).

[antoniopedrofalcaolopesmor]

It simplifies the calculations if you would rationalise 1/phi right at the beginning:
phi+1/phi=
=(1+sqrt5)/2+2(1-sqrt5)/(-4)=
=(1+sqrt5)/2-(1-sqrt5)/2=
=(1+sqrt5-1+sqrt5)/2=
=sqrt5

[Crafty_Art]

as 1/phi = phi - 1
so phi + I/phi = phi + phi - 1
= 2* [(1+-sqrt5)/2] - 1
= +- sqrt5
It's that simple

[antoniopedrofalcaolopesmor]

The shortest path is to use
1/phi=phi-1.
Plugging in the given expression we get:
phi+1/phi=
phi+phi-1=
=2phi-1=
=1+sqrt5-1=
=sqrt5, and we're done!
The proof of the identity at the beginning is easy:
We know that
phi^2-phi-1=0.
Divide both sides by phi
phi-1-1/phi=0
1/phi=phi-1.