Answer:
So, you want to find a number xx such that x2=x+1x2=x+1 ?
Well, that’s a fun exercise. This is a quadratic equation, so we’ll have to learn how to solve those.
First, let’s re-arrange it a bit:
x2−x−1=0x2−x−1=0
(I subtracted x+1x+1 from both sides of the equation, which is always allowed.)
Do you remember that (a−b)2=a2−2ab+b2(a−b)2=a2−2ab+b2 ? Well, we’re going to use that, but we will use it backward. We want to write our equation as (x−b)2−k=0(x−b)2−k=0 , where bb and kk are some values we do not know yet.
Well, if (x−b)2−k=x2−x−1(x−b)2−k=x2−x−1 , and we want to find bb and kk , we can expand the first bit (using the formula above) and get:
x2−2bx+b2−k=x2−x−1x2−2bx+b2−k=x2−x−1
We immediately see that we can subtract x2x2 from both sides:
−2bx+b2−k=−x−1−2bx+b2−k=−x−1
We want both sides to be the same, no matter what value xx has. That means that the only solution is that b=12b=12 , and that means that k=54k=54 (you can try to work that out yourself).
Excellent! We now know that the number xx we are looking for must also be a solution of:
(x−12)2−54=0(x−12)2−54=0
We can add 5454 to both sides:
(x−12)2=54(x−12)2=54
For the next step, we can take the positive and negative square roots:
x−12=5√2x−12=52
and
x−12=−5√2x−12=−52
which gives us the two solutions:
x=1+5√2x=1+52
x=1−5√2x=1−52
The first of these two numbers is quite famous. It is approximately 1.6181.618 , and known as the “golden ratio”. It appears a lot in mathematics, since is is the solution of one of the “simplest” quadratic equations.
I hope you get it..
and don't forget to thank me bye bye
#CarryOnLearning
