Step by step
The question
Question: solve 4x⁴ − 17x² + 4 = 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: 4u² − 17u + 4 = 0.
We solve it like before.
3 Find two numbers
There is a number in front of u², but it can still factor.
The number in front of u² is 4.
So we multiply 4 by 4: that is 16.
We need two numbers that multiply to 16 and add to −17.
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 16, smallest first.
We add each pair.
We need 17.
1 and 16 give 17.
Yes!
With the signs: −1 and −16.
4 Split the middle
We write −17u as −u − 16u.
It is the same amount.
5 Group in pairs
We put the first two together, and the last two together.
From the first pair we take out u.
From the second pair we take out −4.
Both leave the same bracket.
Now both parts have 4u − 1 in them.
6 Same bracket out
4u − 1 is in both parts, so we take it out once.
What is left, u − 4, goes in the other bracket.
So we get (4u − 1)(u − 4).
7 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 4u − 1 = 0, we add 1 on both sides.
In u − 4 = 0, we add 4 on both sides.
Now we have 4u = 1 or u = 4.
To remove times 4, we divide both sides by 4.
So u = 1/4 or u = 4.
Careful!
We found u, not x!
u is x².
One more step to get x.
8 Back to x
u is x².
So x² = 1/4, 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.
1/2 times 1/2 is 1/4.
So x is plus or minus 1/2.
2 times 2 is 4.
So x is plus or minus 2.
So the answers are −2, −1/2, 1/2 and 2.
9 Check
Let's check: put x = 2 in.
−2 gives the same, because the powers are even.
Now put x = 1/2 in.
−1/2 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: find two numbers.
Then: split the middle.
Then: group in pairs.
Then: same bracket out.
Then: one bracket is 0.
Then: back to x.
Last: check.