Step by step
The question
Question: solve √(x + 13) = 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 + 13 = x² + 2x + 1.
2 Get 0 on one side
We take x + 13 away from both sides.
Then the left side is 0.
Like terms together: 0 = x² + x − 12.
3 Factor
We need two numbers.
They multiply to −12, 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 12, smallest first.
We take the smaller from the bigger.
We need 1.
3 and 4 give 1.
Yes!
With the signs: −3 and 4.
So it is 0 = (x − 3)(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 − 3 = 0 gives x = 3.
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 = 3 in the question.
The left side is 4.
The right side is 4.
The same!
x = 3 works.
Now x = −4.
The left side is 3.
The right side is −3.
Not the same!
x = −4 is a false answer.
Books call it an extraneous answer.
We cross out the false answer.
So only x = 3 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.