Worked examples
The question
Question: solve x² = 25.
x2 = 25
1 Square root of both sides
x squared is 25.
So x is the square root of 25, or minus it.
Minus times minus is plus, so the negative one works too.
x² = k → x = √k or x = −√k
5 × 5 = 25, (−5) × (−5) = 25
5 times 5 is 25.
So x = 5 or x = −5.
x = 5 or x = −5
Careful!
Many people write only x = 5.
They lose the minus answer.
Careful: not only x = 5 ✗
2 Check
Let's check: put x = 5 into the left side.
x = 5: 52 = 25
We get 25, the same as the right side.
It works!
Now put x = −5 into the left side.
x = −5: (−5)2 = 25
We get 25 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 + 6x = 7.
x2 + 6x = 7
1 Get 0 on one side
To solve it, we want 0 on the right side.
So we take away 7 on both sides.
We do it on both sides, so they stay equal.
x2 + 6x − 7 = 7 − 7
Now we have x² + 6x − 7 = 0.
x2 + 6x − 7 = 0
2 Factor
We need two numbers.
They multiply to −7, and they add to 6.
? × ? = −7
? + ? = 6
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 7, smallest first.
We take the smaller from the bigger.
We need 6.
1 and 7 give 6.
Yes!
With the signs: −1 and 7.
1 × 7: 7 − 1 = 6 ✓
So it is (x − 1)(x + 7) = 0.
(x − 1)(x + 7) = 0
3 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
We make each bracket 0.
Two small equations.
x − 1 = 0 or x + 7 = 0
In x − 1 = 0, we add 1 on both sides.
In x + 7 = 0, we take away 7 on both sides.
x − 1 + 1 = 0 + 1 or x + 7 − 7 = 0 − 7
So x = 1 or x = −7.
x = 1 or x = −7
4 Check
Let's check: put x = 1 into the left side.
x = 1: 12 + 6(1) = 1 + 6 = 7
We get 7, the same as the right side.
It works!
Now put x = −7 into the left side.
x = −7: (−7)2 + 6(−7) = 49 − 42 = 7
We get 7 again, the same as the right side.
So both answers are right!
Let's look back.
First: get 0 on one side.
Then: factor.
Then: one bracket is 0.
Last: check.
▶ Watch it solved
The question
Question: solve 2x² − 7x + 3 = 0.
2x2 − 7x + 3 = 0
1 Find two numbers
There is a number in front of x², but it can still factor.
The number in front of x² is 2.
So we multiply 2 by 3: that is 6.
We need two numbers that multiply to 6 and add to −7.
2 × 3 = 6
? × ? = 6, ? + ? = −7
They multiply to a plus number, so the two numbers have the same sign.
They add to a minus number, so both are minus.
× is + → same signs: both −
We list the pairs that make 6, smallest first.
We add each pair.
We need 7.
1 and 6 give 7.
Yes!
With the signs: −1 and −6.
1 × 6: 1 + 6 = 7 ✓
2 Split the middle
We write −7x as −x − 6x.
It is the same amount.
2x2 − x − 6x + 3 = 0
3 Group in pairs
We put the first two together, and the last two together.
(2x2 − x) + (−6x + 3) = 0
From the first pair we take out x.
From the second pair we take out −3.
Both leave the same bracket.
(2x2 − x) = x(2x − 1)
(−6x + 3) = −3(2x − 1)
Now both parts have 2x − 1 in them.
x(2x − 1) − 3(2x − 1) = 0
4 Same bracket out
2x − 1 is in both parts, so we take it out once.
What is left, x − 3, goes in the other bracket.
ab + cb = b(a + c)
So we get (2x − 1)(x − 3).
(2x − 1)(x − 3) = 0
5 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
We make each bracket 0.
Two small equations.
2x − 1 = 0 or x − 3 = 0
In 2x − 1 = 0, we add 1 on both sides.
In x − 3 = 0, we add 3 on both sides.
2x − 1 + 1 = 0 + 1 or x − 3 + 3 = 0 + 3
Now we have 2x = 1 or x = 3.
2x = 1 or x = 3
To remove times 2, we divide both sides by 2.
So x = 1/2 or x = 3.
x = 12 or x = 3
6 Check
Let's check: put x = 1/2 into the left side.
x = 12: 2(12)2 − 7(12) + 3 = 12 − 72 + 3 = 0
We get 0, the same as the right side.
It works!
Now put x = 3 into the left side.
x = 3: 2(3)2 − 7(3) + 3 = 18 − 21 + 3 = 0
We get 0 again, the same as the right side.
So both answers are right!
Let's look back.
First: find two numbers.
Then: split the middle.
Then: group in pairs.
Then: same bracket out.
Then: one bracket is 0.
Last: check.
▶ Watch it solved
