Step by step
The question
Question: solve √(x + 7) = x + 1.
x is under a square root.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Square both sides
We want x out from under the square root.
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 + 7 = x² + 2x + 1.
2 Get 0 on one side
Now there is an x².
For x², we get 0 on one side, then factor.
Factor means: write it as a times, with brackets.
We take x + 7 away from both sides.
x + 7 take away x + 7 is 0.
So the left side is 0.
Like terms together.
So 0 = x² + x − 6.
3 Factor
We want two brackets: (x + ?)(x + ?).
We need two numbers.
They multiply to −6, and they add to 1.
x alone means 1x, so the middle number is 1.
Opening the brackets at the end will show why.
Look at the sign in front of 6: it is minus.
So one number is plus and one is minus.
Now the sign in front of x: it is plus.
So the bigger one is plus.
We list the pairs that make 6, smallest first.
For each pair, we check two things: they multiply to 6, and the bigger take away the smaller is 1.
1 × 6 = 6.
6 − 1 = 5.
Not 1.
2 × 3 = 6.
3 − 2 = 1.
Both work!
So the two numbers are 2 and 3.
With the signs: −2 and 3.
The two numbers go in the brackets.
Why these two?
We open the brackets to see.
−2 times 3 makes the −6.
−2x and 3x together make the x.
So it is 0 = (x − 2)(x + 3).
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.
To remove the − 2 from x − 2, we add 2.
To remove the + 3 from x + 3, we take away 3.
x − 2 = 0 gives x = 2.
x + 3 = 0 gives 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 = 2 in the question.
The left side is 3.
The right side is 3.
The same!
x = 2 works.
Now x = −3.
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.
Why did it come?
Squared, −2 and 2 both give 4.
Squaring hid the minus.
We cross out the false answer.
So only x = 2 is an answer.
6 Pick the choice
Our answer is x = 2.
That is choice C.
7 Why not the others
A is −3.
Wrong: it does not check: √4 is 2, but −3 + 1 is −2.
B is −3 and 2.
Wrong: −3 does not check: √4 is 2, but −3 + 1 is −2.
D is 3.
Wrong: squared x + 1 as x² + 1: forgot the 2x.
Let's look back.
First: square both sides.
Then: get 0 on one side.
Then: factor.
Then: one bracket is 0.
Then: check.
Then: pick the choice.
Last: why not the others.