Step by step
The question
Question: solve √(x + 1) = x − 5.
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 − 5) squared is (x − 5) times (x − 5).
Each part times each part.
Then the like terms: x² − 10x + 25.
Careful!
It is not x squared plus 25.
The middle part −10x is there too.
Now we have x + 1 = x² − 10x + 25.
2 Get 0 on one side
We take x + 1 away from both sides.
Then the left side is 0.
Like terms together: 0 = x² − 11x + 24.
3 Factor
We need two numbers.
They multiply to 24, and they add to −11.
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 24, smallest first.
We add each pair.
We need 11.
3 and 8 give 11.
Yes!
With the signs: −3 and −8.
So it is 0 = (x − 3)(x − 8).
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 − 3 = 0 gives x = 3.
x − 8 = 0 gives x = 8.
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 = 3 in the question.
The left side is 2.
The right side is −2.
Not the same!
x = 3 is a false answer.
Books call it an extraneous answer.
Now x = 8.
The left side is 3.
The right side is 3.
The same!
x = 8 works.
We cross out the false answer.
So only x = 8 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.