Worked examples
The question
Question: simplify (2x + 3) + (x − 5).
Simplify means: write it shorter, with the same value.
(2x + 3) + (x − 5)
1 Open the brackets
A bracket with a plus or nothing in front can just be removed.
( ) away
Now we have 2x + 3 + x − 5.
= 2x + 3 + x − 5
2 Group like terms
Terms are the parts between the + and − signs.
Like terms are the same kind: same letters, same powers.
All the x terms go together.
Plain numbers go together.
Each colour is one kind.
like terms = same letters,
same powers
We move like terms next to each other.
Each term takes its sign with it.
= 2x + x + 3 − 5
2x + x is 3x.
It is like 2 apples plus 1 apple: that is 3 apples.
3 − 5 is −2.
2x + x = 3x
3 − 5 = −2
So the expression is 3x − 2.
= 3x − 2
3 Check
Let's check.
We pick a number for each letter: x = 2.
We put it into the start.
Start: (2 × 2 + 3) + (2 − 5) = 7 + (−3)
= 7 − 3 = 4
Now into our answer.
Answer: 3 × 2 − 2 = 6 − 2 = 4
Both give 4.
Same value, so 3x − 2 is right!
Both give 4
Let's look back.
First: open the brackets.
Then: group like terms.
Last: check.
▶ Watch it solved
The question
Question: simplify (3x² + 2x − 1) − (x² − 5x + 4).
Simplify means: write it shorter, with the same value.
(3x2 + 2x − 1) − (x2 − 5x + 4)
1 Open the brackets
A bracket with a plus or nothing in front can just be removed.
A minus in front of a bracket changes the sign of every part inside.
−( + ) → −, −( − ) → +
Now we have 3x² + 2x − 1 − x² + 5x − 4.
= 3x2 + 2x − 1 − x2 + 5x − 4
2 Group like terms
Terms are the parts between the + and − signs.
Like terms are the same kind: same letters, same powers.
All the x² terms go together.
Plain numbers go together.
Each colour is one kind.
like terms = same letters,
same powers
We move like terms next to each other.
Each term takes its sign with it.
= 3x2 − x2 + 2x + 5x − 1 − 4
3x² − x² is 2x².
It is like 3 apples take away 1 apple: that is 2 apples.
2x + 5x is 7x.
−1 − 4 is −5.
3x2 − x2 = 2x2
2x + 5x = 7x
−1 − 4 = −5
So the expression is 2x² + 7x − 5.
= 2x2 + 7x − 5
3 Check
Let's check.
We pick a number for each letter: x = 2.
We put it into the start.
Start: (3 × 22 + 2 × 2 − 1) − (22 − 5 × 2 + 4)
= 15 − (−2) = 15 + 2 = 17
Now into our answer.
Answer: 2 × 22 + 7 × 2 − 5 = 8 + 14 − 5
= 17
Both give 17.
Same value, so 2x² + 7x − 5 is right!
Both give 17
Let's look back.
First: open the brackets.
Then: group like terms.
Last: check.
▶ Watch it solved
The question
Question: simplify (3x − 2)(2x + 5).
Simplify means: write it shorter, with the same value.
(3x − 2)(2x + 5)
1 Split the first bracket
Each part of the first bracket multiplies the whole second bracket.
(a + b)(c + d) = a(c + d) + b(c + d)
The parts of the first bracket are 3x and −2.
Each one takes the whole second bracket.
= 3x(2x + 5) − 2(2x + 5)
2 Open the brackets
What is in front of a bracket multiplies every part inside.
a(b + c) = ab + ac
So we write each part times what is in front.
= 3x(2x) + 3x(5) + (−2)(2x) + (−2)(5)
− × + = −
Watch the signs.
3x times 2x is 6x².
3x times 5 is 15x.
(−2) times 2x is −4x.
(−2) times 5 is −10.
= 6x2 + 15x − 4x − 10
3 Group like terms
Terms are the parts between the + and − signs.
Like terms are the same kind: same letters, same powers.
All the x terms go together.
Each colour is one kind.
like terms = same letters,
same powers
15x − 4x is 11x.
It is like 15 apples take away 4 apples: that is 11 apples.
15x − 4x = 11x
So the expression is 6x² + 11x − 10.
= 6x2 + 11x − 10
4 Check
Let's check.
We pick a number for each letter: x = 2.
We put it into the start.
Start: (3 × 2 − 2)(2 × 2 + 5) = (4)(9) = 36
Now into our answer.
Answer: 6 × 22 + 11 × 2 − 10
= 24 + 22 − 10 = 36
Both give 36.
Same value, so 6x² + 11x − 10 is right!
Both give 36
Let's look back.
First: split the first bracket.
Then: open the brackets.
Then: group like terms.
Last: check.
▶ Watch it solved
