Step by step
The question
Question: solve √(3x + 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.
The squares remove both square roots.
2 Get x on one side
There is x on both sides.
We want x on one side only.
So we remove the x from the right side.
To remove x, we take away x.
We do it on both sides, so they stay equal.
x and − x make zero.
Cross them out.
3x − x = 2x.
Now we have 2x + 1 = 5.
We want x by itself.
Two things are stuck to x: times 2 and + 1.
The + 1 is on the outside, so it comes off first.
Then the times 2.
3 Remove the + 1
To remove + 1, we take away 1.
We do it on both sides, so they stay equal.
+ 1 and − 1 make zero.
Cross them out.
5 − 1 = 4.
Now we have 2x = 4.
4 Remove the 2
2x means 2 times x.
We want just x.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
So just x is left.
4 divided by 2 is 2.
So x = 2!
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 = 2 in the question.
The left side is √7.
The right side is √7.
The same!
x = 2 works.
Let's look back.
First: square both sides.
Then: get x on one side.
Then: remove the + 1.
Then: remove the 2.
Last: check.