Step by step
The question
Question: solve x − 7√x + 12 = 0.
1 Spot the pattern
x is the square of √x: √x times √x is x.
So the equation has √x squared, and √x, and a number.
It looks like a quadratic.
2 Name u = √x
We give √x a short name: u.
Then √x squared is u squared.
Now it is a quadratic in u: u² − 7u + 12 = 0.
We solve it like before.
3 Factor
We need two numbers.
They multiply to 12, and they add to −7.
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 12, smallest first.
We add each pair.
We need 7.
3 and 4 give 7.
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 add 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 √x.
One more step to get x.
5 Back to x
u is √x.
So √x = 3, or √x = 4.
To remove the square root, we square both sides: 3 squared is 9.
To remove the square root, we square both sides: 4 squared is 16.
So the answers are 9 and 16.
6 Check
Let's check: put x = 9 in.
Now put x = 16 in.
Every answer gives 0.
They all work!
Let's look back.
First: spot the pattern.
Then: name u = √x.
Then: factor.
Then: one bracket is 0.
Then: back to x.
Last: check.