Step by step
The question
Question: solve x⁴ − 10x² + 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 − 10u + 9 = 0.
We solve it like before.
3 Factor
We need two numbers.
They multiply to 9, and they add to −10.
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 9, smallest first.
We add each pair.
We need 10.
1 and 9 give 10.
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 add 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.
1 times 1 is 1.
So x is plus or minus 1.
3 times 3 is 9.
So x is plus or minus 3.
So the answers are −3, −1, 1 and 3.
6 Check
Let's check: put x = 3 in.
−3 gives the same, because the powers are even.
Now put x = 1 in.
−1 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.