Step by step
The question
Question: solve x⁶ − 26x³ − 27 = 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 − 26u − 27 = 0.
We solve it like before.
3 Factor
We need two numbers.
They multiply to −27, and they add to −26.
They multiply to a minus number, so one number is plus and one is minus.
They add to a minus number, so the bigger one is minus.
We list the pairs that make 27, smallest first.
We take the smaller from the bigger.
We need 26.
1 and 27 give 26.
Yes!
With the signs: 1 and −27.
So it is (u + 1)(u − 27) = 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 take away 1 on both sides.
In u − 27 = 0, we add 27 on both sides.
So u = −1 or u = 27.
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 = 27.
−1 times −1 times −1 is −1.
A cube root has just one answer: x = −1.
3 times 3 times 3 is 27.
A cube root has just one answer: x = 3.
So the answers are −1 and 3.
6 Check
Let's check: put x = −1 in.
Now put x = 3 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.