The question
Question: solve √x = 5.
√x = 5
1 Square both sides
Squaring removes the square root.
The square root of 9 is 3, and 3 squared is 9 again.
(√9)² = 9
We square both sides, so they stay equal.
(√x)2 = 52
The square removes the square root.
5 squared is 25.
x = 25
So x = 25!
2 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.
Careful: squaring can make a false answer
(−2)² = 4 and 2² = 4
Let's check x = 25 in the question.
The left side is 5.
x = 25: √25 = 5
The right side is 5.
The same!
x = 25 works.
right side: 5 ✓
Let's look back.
First: square both sides.
Last: check.
▶ Watch it solved
The question
Question: solve √(2x − 1) + 3 = 6.
√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.
√2x − 1 + 3 − 3 = 6 − 3
+ 3 and − 3 make zero.
Cross them out.
6 minus 3 is 3.
6 − 3 = 3
Now we have √(2x − 1) = 3.
√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.
(√9)² = 9
We square both sides, so they stay equal.
(√2x − 1)2 = 32
The square removes the square root.
3 squared is 9.
2x − 1 = 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.
2x − 1 + 1 = 9 + 1
− 1 and + 1 make zero.
Cross them out.
9 + 1 = 10.
9 + 1 = 10
Now we have 2x = 10.
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.
2x2 = 102
2 divided by 2 is 1.
So just x is left.
10 divided by 2 is 5.
102 = 5
So x = 5!
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.
Careful: squaring can make a false answer
(−2)² = 4 and 2² = 4
Let's check x = 5 in the question.
The left side is 6.
x = 5: √2(5) − 1 + 3 = √9 + 3 = 3 + 3 = 6
The right side is 6.
The same!
x = 5 works.
right side: 6 ✓
Let's look back.
First: get the root alone.
Then: square both sides.
Then: remove the − 1.
Then: remove the 2.
Last: check.
▶ Watch it solved
The question
Question: solve √(x + 7) = x + 1.
√x + 7 = x + 1
1 Square both sides
Squaring removes the square root.
The square root of 9 is 3, and 3 squared is 9 again.
(√9)² = 9
We square both sides, so they stay equal.
(√x + 7)2 = (x + 1)2
(x + 1) squared is (x + 1) times (x + 1).
Each part times each part.
Then the like terms: x² + 2x + 1.
(x + 1)² = (x + 1)(x + 1)
= x² + x + x + 1
= x² + 2x + 1
Careful!
It is not x squared plus 1.
The middle part 2x is there too.
Careful: (x + 1)² is not x² + 1
Now we have x + 7 = x² + 2x + 1.
x + 7 = x2 + 2x + 1
2 Get 0 on one side
We take x + 7 away from both sides.
Then the left side is 0.
0 = x2 + 2x + 1 − x − 7
Like terms together: 0 = x² + x − 6.
0 = x2 + x − 6
3 Factor
We need two numbers.
They multiply to −6, and they add to 1.
? × ? = −6
? + ? = 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.
× is − → one +, one −; bigger one +
We list the pairs that make 6, smallest first.
We take the smaller from the bigger.
We need 1.
2 and 3 give 1.
Yes!
With the signs: −2 and 3.
1 × 6: 6 − 1 = 5
2 × 3: 3 − 2 = 1 ✓
So it is 0 = (x − 2)(x + 3).
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.
A × B = 0 → A = 0 or B = 0
x − 2 = 0 gives x = 2.
x + 3 = 0 gives x = −3.
x = 2 or 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.
Careful: squaring can make a false answer
(−2)² = 4 and 2² = 4
Let's check x = 2 in the question.
The left side is 3.
x = 2: √2 + 7 = √9 = 3
The right side is 3.
The same!
x = 2 works.
right side: 2 + 1 = 3 ✓
Now x = −3.
The left side is 2.
x = −3: √−3 + 7 = √4 = 2
The right side is −2.
Not the same!
x = −3 is a false answer.
Books call it an extraneous answer.
right side: −3 + 1 = −2 ≠ 2
We cross out the false answer.
So only x = 2 is an answer.
x = 2
Let's look back.
First: square both sides.
Then: get 0 on one side.
Then: factor.
Then: one bracket is 0.
Last: check.
▶ Watch it solved
