Worked examples
The question
Question: solve x² = 49.
x2 = 49
1 Square root of both sides
x squared is 49.
So x is the square root of 49, or minus it.
Minus times minus is plus, so the negative one works too.
x² = k → x = √k or x = −√k
7 × 7 = 49, (−7) × (−7) = 49
7 times 7 is 49.
So x = 7 or x = −7.
x = 7 or x = −7
Careful!
Many people write only x = 7.
They lose the minus answer.
Careful: not only x = 7 ✗
2 Check
Let's check: put x = 7 into the left side.
x = 7: 72 = 49
We get 49, the same as the right side.
It works!
Now put x = −7 into the left side.
x = −7: (−7)2 = 49
We get 49 again, the same as the right side.
So both answers are right!
Let's look back.
First: square root of both sides.
Last: check.
▶ Watch it solved
The question
Question: solve x² = 144.
x2 = 144
1 Square root of both sides
x squared is 144.
So x is the square root of 144, or minus it.
Minus times minus is plus, so the negative one works too.
x² = k → x = √k or x = −√k
12 × 12 = 144, (−12) × (−12) = 144
12 times 12 is 144.
So x = 12 or x = −12.
x = 12 or x = −12
Careful!
Many people write only x = 12.
They lose the minus answer.
Careful: not only x = 12 ✗
2 Check
Let's check: put x = 12 into the left side.
x = 12: 122 = 144
We get 144, the same as the right side.
It works!
Now put x = −12 into the left side.
x = −12: (−12)2 = 144
We get 144 again, the same as the right side.
So both answers are right!
Let's look back.
First: square root of both sides.
Last: check.
▶ Watch it solved
The question
Question: solve x³ = 20.
x3 = 20
x³ means x times x times x.
We want just x.
x³ = x × x × x
1 Cube root of both sides
The cube root asks: which number, times itself three times, makes this?
∛27 = ? → ? × ? × ? = 27
The cubes: 1 times 1 times 1 is 1.
2 times 2 times 2 is 8.
3 times 3 times 3 is 27.
1³ = 1, 2³ = 8, 3³ = 27
x is the cube root of 20.
x = 3√20
20 is not a cube.
It is between 8 and 27.
So the answer is between 2 and 3.
2³ = 8 < 20 < 27 = 3³
A calculator gives 2.714, and so on.
x = 2.714…
To the nearest tenth, x is about 2.7.
x ≈ 2.7
Careful!
A cube has one answer, not two.
A negative number cubed is negative.
Not like x² = 49.
Careful: not x ≈ ±2.7 ✗
2 Check
Let's check: 2.7 times 2.7 times 2.7 is 19.683.
Very close to 20: we rounded.
So the answer is right!
2.7 × 2.7 × 2.7 = 19.683 ≈ 20
Let's look back.
First: cube root of both sides.
Last: check.
Answer Careful: not x ≈ ±2.7 ✗
▶ Watch it solved
