Step by step
The question
Question: solve √(2x − 1) + 3 = 6.
1 Get the root alone
We want the square root by itself.
The + 3 is in the way.
To remove + 3, we take away 3.
We do it on both sides, so they stay equal.
+ 3 and − 3 make zero.
Cross them out.
6 minus 3 is 3.
Now we have √(2x − 1) = 3.
2 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 square removes the square root.
3 squared is 9.
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 add 1.
We do it on both sides, so they stay equal.
− 1 and + 1 make zero.
Cross them out.
9 + 1 = 10.
Now we have 2x = 10.
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.
10 divided by 2 is 5.
So x = 5!
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 = 5 in the question.
The left side is 6.
The right side is 6.
The same!
x = 5 works.
Let's look back.
First: get the root alone.
Then: square both sides.
Then: remove the − 1.
Then: remove the 2.
Last: check.