Step by step
The question
Question: solve x⁴ − x² − 12 = 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 − u − 12 = 0.
We solve it like before.
3 Factor
We need two numbers.
They multiply to −12, and they add to −1.
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 12, smallest first.
We take the smaller from the bigger.
We need 1.
3 and 4 give 1.
Yes!
With the signs: 3 and −4.
So it is (u + 3)(u − 4) = 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 + 3 = 0, we take away 3 on both sides.
In u − 4 = 0, we add 4 on both sides.
So u = −3 or u = 4.
Careful!
We found u, not x!
u is x².
One more step to get x.
5 Back to x
u is x².
So x² = −3, or x² = 4.
x squared is a number.
So x is its square root, or minus it.
Both work, because minus times minus is plus.
x² = −3: no real number squared is negative.
2 times 2 is 4.
So x is plus or minus 2.
So the answers are −2 and 2.
6 Check
Let's check: put x = 2 in.
−2 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.