Step by step
The question
Question: solve x⁶ − 9x³ + 8 = 0.
1 Spot the pattern
x⁶ is x³ times x³, so x⁶ is x³ squared.
So the equation has x³ squared, and x³, and a number.
It looks like a quadratic.
2 Name u = x³
We give x³ a short name: u.
Then x³ squared is u squared.
Now it is a quadratic in u: u² − 9u + 8 = 0.
We solve it like before.
3 Factor
We need two numbers.
They multiply to 8, and they add to −9.
They multiply to a plus number, so the two numbers have the same sign.
They add to a minus number, so both are minus.
We list the pairs that make 8, smallest first.
We add each pair.
We need 9.
1 and 8 give 9.
Yes!
With the signs: −1 and −8.
So it is (u − 1)(u − 8) = 0.
4 One bracket is 0
Two numbers multiply to 0.
That only happens if one of them is 0.
So one bracket must be 0.
We make each bracket 0.
Two small equations.
In u − 1 = 0, we add 1 on both sides.
In u − 8 = 0, we add 8 on both sides.
So u = 1 or u = 8.
Careful!
We found u, not x!
u is x³.
One more step to get x.
5 Back to x
u is x³.
So x³ = 1, or x³ = 8.
1 times 1 times 1 is 1.
A cube root has just one answer: x = 1.
2 times 2 times 2 is 8.
A cube root has just one answer: x = 2.
So the answers are 1 and 2.
6 Check
Let's check: put x = 1 in.
Now put x = 2 in.
Every answer gives 0.
They all work!
Let's look back.
First: spot the pattern.
Then: name u = x³.
Then: factor.
Then: one bracket is 0.
Then: back to x.
Last: check.