– English Lyrics

Lyrics & Translation

[English]

Today we will be solving a nice math

olympiad question which is if x + y = 4

and x to the power 5 + y to the^ 5 =

464. Find out the values of x and y.

This problem looks hard but there's a

neat way to solve it. Let us start. We

all know that x + y square is x^2 + 2 *

xy + y 2. Right? So take this 2 * xy

this side to get x 2 + y 2 as x + y^ 2 -

2 * x y. But x + y is 4. And thus we get

x^2 + y^ 2 = 4^ 2 - 2 * x y which is 16

- 2 * x y. Then we might also know the

formula for x + y whole cube. It's x cub

+ yub + 3 * xy * x + y. So substitute x

+ y as 4 to get 4 cub = x cub + yub + 3

xy * 4. This becomes 64 = x cub + y cub

+ 12 x y. Thus x cub + y cub = 64 - 12 x

y. But now what about x + y whole raised

to power 4 and other higher powers? How

to expand them? To do so we rewrite this

four as 1 + 3. Now according to the

power law a raised to m + n = a raised

to m * a raised to power n. So we can

rewrite this as x + y raised to the

power 1 which is nothing but x + y * x +

y raised to the power 3. That gives us a

way to break down higher powers using

lower ones we already know. Substitute x

+ y whole cube as this. Now let us

expand it. Take this and multiply it

with x + y to get this. Then take this

and multiply it with x + y to get 3 xy *

x + y square. Now let us expand this

part. x * x cub is x 4th. Then this and

this becomes x * yub. Then this and this

becomes y * x cub and y * y cub will be

y 4. So this will become x 4 + y raised

to 4 + this. So take x y as common from

them to get xy * x^2 + y

2. Now substitute x^2 + y 2 from here to

get this as xy * 16 - 2 x y. So this

becomes x^ 4 + y 4 + 16 xy - 2 * xy

whole square. But don't forget we also

have this which is 3 * xy * this will

become 4 square or 16. Thus this will be

48 * x y. So x + y whole raised to 4

becomes x to the 4 + y to the 4 + both

these will add up to give 64 xy - 2 * xy

2. So finally we are ready to expand x +

y whole raised to power 5 which can be

rewritten as x + y * x + y whole raised

to 4 which is this. So substitute it

here. Now take this x + y from here and

x to the 4th power + y to the 4th power

and multiply them. Then take this and

multiply it with x + y. First let us

expand the easier part which is this.

Simply put x = 4 here to get this entire

thing as 4 * this which will be 4 * 64

or

256xy - 8 xy square. Now let us expand

this part. we get x * x 4 or x 5 + this

and this will become x * y 4 and this

will become y * x 4 + y * y to the 4

gives y 5. So we have x 5 + y 5 plus

take x y as common from here to get xy *

x cub + y cub. Now substitute x to the 5

+ y to the 5 as 464 and x cub + y cub as

64 - 12 xy to get this part as 464 +

64xy -2 xy 2 and then we also have this

part which is 256 xy - 8xy square. So it

becomes 464 plus this and this will give

320 xy and this and this will give -

20xy whole square which = x + y whole 5.

Substitute 4 here to get 4 to 5 which

will be equal to 1,24.

So take everything on left side to get

this and finally we get 20 xy^ 2 - 320

xy + 560 = 0. Let us assume xy = some

variable t and then also divide by 20 on

all sides to get this as t ^2 - 16 t +

28 = 0. You know I will not bore you by

solving this quadratic equation. we get

t as 2 or t = 14 which means xy = 2 or

xy = 14. Now we have xy = t and thus we

get y = t /x. Substitute y in x + y = 4

to get x + t /x = 4. Multiply by x on

all sides to get x^2 + t = 4x. So x^2 -

4x + t = 0. Since there are two values

of t. Thus we get these two quadratic

equations. Again use this quadratic

formula to solve for x and we get x = 2

+ <unk>2 and 2 -<unk>2 for this

equation. Then we get 2 + i<unk>10 and 2

- i<unk> 10 for this equation where i

equ= the square root of the 1. This

means it gives a complex root. Now

finding y is super easy. We have x + y =

4 which means y = 4 - x. So for this

value of x we get y as 4 - 2 +

square<unk> 2 or 2 -<unk>2. Then for

this value of x we get y as 4 - 2 -

<unk>2 or 2 + <unk>2. Next for this

value of x we get y as 4 - 2 + i<unk> 10

or 2 - i<unk> 10. Then finally for this

value of x we get y as 4 - 2 - i<unk> 10

or 2 + i<unk> 10. This way we get four

sets of values of x and y. That was

simply out of this world. If you enjoyed

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good day.

[Music]