Step by step
The question
Question: solve √(x + 5) = x − 1.
1 Square both sides
Squaring removes the square root.
The square root of 9 is 3, and 3 squared is 9 again.
We square both sides, so they stay equal.
(x − 1) squared is (x − 1) times (x − 1).
Each part times each part.
Then the like terms: x² − 2x + 1.
Careful!
It is not x squared plus 1.
The middle part −2x is there too.
Now we have x + 5 = x² − 2x + 1.
2 Get 0 on one side
We take x + 5 away from both sides.
Then the left side is 0.
Like terms together: 0 = x² − 3x − 4.
3 Factor
We need two numbers.
They multiply to −4, and they add to −3.
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 4, smallest first.
We take the smaller from the bigger.
We need 3.
1 and 4 give 3.
Yes!
With the signs: 1 and −4.
So it is 0 = (x + 1)(x − 4).
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.
x + 1 = 0 gives x = −1.
x − 4 = 0 gives x = 4.
5 Check
Careful!
Squaring can make an answer that does not work.
Why?
Minus 2 squared is 4, and 2 squared is 4 too.
Squaring hides a minus sign.
So we check every answer in the question.
Let's check x = −1 in the question.
The left side is 2.
The right side is −2.
Not the same!
x = −1 is a false answer.
Books call it an extraneous answer.
Now x = 4.
The left side is 3.
The right side is 3.
The same!
x = 4 works.
We cross out the false answer.
So only x = 4 is an answer.
Let's look back.
First: square both sides.
Then: get 0 on one side.
Then: factor.
Then: one bracket is 0.
Last: check.