Step by step
The question
Question: solve √(4x + 1) = √(2x + 7).
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 2x from the right side.
To remove 2x, we take away 2x.
We do it on both sides, so they stay equal.
2x and − 2x make zero.
Cross them out.
4x − 2x = 2x.
Now we have 2x + 1 = 7.
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.
7 − 1 = 6.
Now we have 2x = 6.
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.
6 divided by 2 is 3.
So x = 3!
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 √13.
The right side is √13.
The same!
x = 3 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.