Step by step
The question
Question: solve x⁴ − 8x² − 9 = 0.
1 Spot the pattern
x⁴ is x² times x², so x⁴ is x² squared.
So the equation has x² squared, and x², and a number.
It looks like a quadratic.
2 Name u = x²
We give x² a short name: u.
Then x² squared is u squared.
Now it is a quadratic in u: u² − 8u − 9 = 0.
We solve it like before.
3 Factor
We need two numbers.
They multiply to −9, and they add to −8.
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 9, smallest first.
We take the smaller from the bigger.
We need 8.
1 and 9 give 8.
Yes!
With the signs: 1 and −9.
So it is (u + 1)(u − 9) = 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 − 9 = 0, we add 9 on both sides.
So u = −1 or u = 9.
Careful!
We found u, not x!
u is x².
One more step to get x.
5 Back to x
u is x².
So x² = −1, or x² = 9.
x squared is a number.
So x is its square root, or minus it.
Both work, because minus times minus is plus.
x² = −1: no real number squared is negative.
3 times 3 is 9.
So x is plus or minus 3.
So the answers are −3 and 3.
6 Check
Let's check: put x = 3 in.
−3 gives the same, because the powers are even.
Every answer gives 0.
They all work!
Let's look back.
First: spot the pattern.
Then: name u = x².
Then: factor.
Then: one bracket is 0.
Then: back to x.
Last: check.