Tunjukkan means prove
#idk how prove this equation
hot gazelle
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Tunjukkan means prove
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hallow comet
Tunjukkan means prove
Let the expression on the left be x. Then we have:
x = (2√(13) + 5)^(1/3) - (2√(13) - 5)^(1/3)
Start by cubing x.
hot gazelle
Then x^3 = 1 ?
hallow comet
No. For now, pretend like you don't know it's 1.
hallow comet
Do you remember the formula for (a - b)^3?
hallow comet
No. Should be a^3 - 3a^2 b + 3ab^2 - b^3.
hallow comet
Before you do that, one more thing.
Let's take a look at that formula.
(a - b)^3 = a^3 - 3a^2 b + 3ab^2 - b^3
Let's group some terms.
(a - b)^3 = (a^3 - b^3) - (3a^2 b - 3ab^2)
So:
(a - b)^3 = (a^3 - b^3) - 3ab(a - b)
hot gazelle
Ohh I see
hallow comet
That will allow you to express x^3 in terms of x.
hot gazelle
Alr I'll cube it first
hallow comet
Let's group some terms.
(a - b)^3 = (a^3 - b^3) - (3a^2 b - 3ab^2)
So:
(a - b)^3...
You can use the grouped variant of the formula right away.
hallow comet
<@290136354233384960>
What did you get?
hallow comet
Well, let's see. We have:
(a - b)^3 = (a^3 - b^3) - 3ab(a - b)
In our case a = (2√(13) + 5)^(1/3) and b = (2√(13) - 5)^(1/3). Let's find a^3 - b^3 and ab.
a^3 - b^3 = (2√(13) + 5) - (2√(13) - 5) = 5 + 5 = 10
ab = ((2√(13) + 5)(2√(13) - 5))^(1/3) = (4*13 - 25)^(1/3) = (52 - 25)^(1/3) = 27^(1/3) = 3
Thus, if a - b = x, we get:
x^3 = 10 - 9x
So, now you just have to prove that x = 1 is the only real root of the function f(x) = x^3 + 9x - 10.
hot gazelle
Well, let's see. We have:
(a - b)^3 = (a^3 - b^3) - 3ab(a - b)
In our case a = (...
Sorry, I still don't understand where the rest of the equation go
Mostly the ab part
hallow comet
Our initial expression is x = a - b. We have (a - b)^3 = (a^3 - b^3) - 3ab(a - b), so x^3 = (a^3 - b^3) - 3abx.
Oh, I used the difference of squares formula there.
hot gazelle
So, now you just have to prove that x = 1 is the only real root of the function ...
Where did you get the x^3?
Is it because 1 = x^3 ?
Nvm I get it
hot gazelle
So, now you just have to prove that x = 1 is the only real root of the function ...
Uhmmm, about this, how do I prove it's 1 ?
hallow comet
Uhmmm, about this, how do I prove it's 1 ?
Well, we got x^3 = 10 - 9x. Obviously, x = 1 is a root of this equation. But we need to determine whether it's the only real root.
Only then we can say that the initial expression is 1.
hallow comet
We already know the solution x = 1.
We just need to show that it's the only real solution.
hot gazelle
Oh
hallow comet
Let's look at the equation we got before.
x^3 = 10 - 9x
What can you say about the left and right sides in terms of whether they increase or decrease?
hot gazelle
They increase ?
hallow comet
Both of them? Are you sure?
hallow comet
Yes! Very good.
And here we can use a useful property.
Suppose we have an equation of the form f(x) = g(x), where one side always increases and one side always decreases. Then if this equation has solutions, then it only has one solution.
hot gazelle
Well the right side decrease and the left side increase
Yk, I said that without knowing which one actually increase or decrease...
Is it because the right side is cube while the left side is not?
hallow comet
Well, x^3 increases and 10 - 9x decreases, obviously.
hot gazelle
Ohhhh
Because it will only add up
While the left one will decrease
So if x was 3, x^3 would be bigger than 10-9x
Ahh i get it
Thx
hot gazelle
And here we can use a useful property.
Suppose we have an equation of the form f...
So then how do I use this on the equation?
hallow comet
So then how do I use this on the equation?
We have:
x^3 = 10 - 9x
We know that x = 1 is its root. But as x^3 is increasing and 10 - 9x is decreasing, this equation has no more than one real root. So, it only has one real root: x = 1.
But x is our initial expression. So, we proved that it is equal to 1.