Step by step
The question
Question: solve x + √x − 6 = 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 + u − 6 = 0.
We solve it like before.
3 Factor
We need two numbers.
They multiply to −6, and they add to 1.
They multiply to a minus number, so one number is plus and one is minus.
They add to a plus number, so the bigger one is plus.
We list the pairs that make 6, smallest first.
We take the smaller from the bigger.
We need 1.
2 and 3 give 1.
Yes!
With the signs: −2 and 3.
So it is (u − 2)(u + 3) = 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 − 2 = 0, we add 2 on both sides.
In u + 3 = 0, we take away 3 on both sides.
So u = 2 or u = −3.
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 = 2.
√x = −3: a square root is never negative.
So this one gives nothing.
To remove the square root, we square both sides: 2 squared is 4.
So the answers are 4.
6 Check
Let's check: put x = 4 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.