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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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