M4.9.1 Chapter 1

Expanding two brackets

I know 3(x + 4) = 3x + 12. What do I do when both parts are brackets, like (x + 3)(x + 2)?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A x² + 6: multiply the first terms, and multiply the last terms.
You picked A. Read on. At the end, the chapter comes back to your guess.
B x² + 5x + 6: multiply every term by every term, then join the like terms.
You picked B. Read on. At the end, the chapter comes back to your guess.
C 2x + 5: add the terms inside the brackets.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Expand (x + 3)(x − 5)
Expand (2x + 1)² without losing the middle term

Amina knows that 3(x + 4) means 3 × x and 3 × 4. The 3 multiplies each part of the bracket. This is called expanding the bracket.

Now take (x + 3)(x + 2). Picture a rectangle that is x + 3 long and x + 2 wide. Its area is the two brackets multiplied together.

The whole rectangle is made of four parts. Its area is x² + 3x + 2x + 6.

The four parts are the four products. x times x is x². x times 2 is 2x. 3 times x is 3x. 3 times 2 is 6. The parts 3x and 2x are alike, so they join: 5x.

Draw it first

Multiply every pair of terms

1 Start with the two brackets, (x + 3)(x + 2).
2 Multiply every term in the first bracket by every term in the second. There are 2 × 2 = 4 products.
3 Join the like terms. 2x + 3x = 5x, so the answer is x² + 5x + 6.

A term with a minus sign needs care. A positive number times a negative number is negative. A negative times a negative is positive. Here is a pattern that shows why.

Going down the table, the second number falls by 1 each time and the answer rises by 5. So −5 × −1 is 5.

A grid keeps the four products in order. Write one bracket across the top and the other down the side. Each cell holds the product of the terms at the ends of its row and column.

The hatched cells are the like terms. Adding them gives 3x − 5x = −2x.

Worked example

Expand (x + 3)(x − 5).

Write the four products: x × x = x², x × (−5) = −5x, 3 × x = 3x and 3 × (−5) = −15.
Add them up: x² − 5x + 3x − 15.
Join the x terms. The numbers in front are −5 and 3, and −5 + 3 = −2.
So the x terms make −2x, and (x + 3)(x − 5) = x² − 2x − 15.

Answer: x² − 2x − 15

You can test an expansion with a number. Put the same value of x into the brackets and into your answer. A right expansion gives the same number both ways.

The first two forms agree. The third gives 23, so it is a wrong expansion.

A bracket squared is the bracket times itself. (2x + 1)² means (2x + 1)(2x + 1). It has four products, not two. The two middle products are equal, and together they make the middle term.

The two hatched middle cells are 2x and 2x. They make 4x.

Worked example

Expand (2x + 1)².

Write the square as two brackets: (2x + 1)(2x + 1).
First times first: 2x × 2x = 4x², because 2 × 2 = 4.
Last times last: 1 × 1 = 1.
The two middle products are 2x × 1 = 2x and 1 × 2x = 2x. Add their numbers: 2 + 2 = 4.
So (2x + 1)² = 4x² + 4x + 1.

Answer: 4x² + 4x + 1

See it three ways

With things

Cut a rectangle that is x + 4 long and x + 2 wide into four parts, and add the parts.

As a picture

In words

Multiply every term in the first bracket by every term in the second. That gives four products. Then join the two like terms, 4x and 2x.

In symbols

(x + 4)(x + 2) = x² + 2x + 4x + 8 = x² + 6x + 8.

Try it

Test an expansion with numbers

You need: paper and a pencil.

Take (x + 3)(x + 2) and its expansion x² + 5x + 6.
Put x = 1 into both. Work out 4 × 3, and then 1 + 5 + 6.
Try x = 2, and then x = 10. Write the three results in a table.

What you will see: The brackets and the expansion give the same number every time. If they ever differ, there is a mistake in the expansion.

If you cannot try it, this is what happens.

A wrong idea many people have

(x + 3)² = x² + 9.

Why it feels right: Squaring one number squares it, so it seems that squaring a bracket squares each part inside.

The test that breaks it: Put x = 1. Then (1 + 3)² = 16, but 1² + 9 = 10. They differ, so x² + 9 is wrong.

What is true: A bracket squared is the bracket times itself, with four products. The two middle terms, 3x and 3x, make 6x. So (x + 3)² = x² + 6x + 9.

A wrong idea many people have

(x + 3)(x − 5) = x² − 2x + 15.

Why it feels right: The last term is 3 times 5, so it feels as if it should be a positive 15.

The test that breaks it: Put x = 4. The brackets give 7 × (−1) = −7. But 4² − 2 × 4 + 15 = 23. The two numbers differ.

What is true: The last product is 3 × (−5), and a positive times a negative is negative. It is −15. Always take the sign with the term.

Key words

expand: to multiply out brackets, so that no brackets are left.
term: one part of an expression that is added or taken away, such as 3x or −15.
like terms: terms with the same letters and powers, such as 3x and −5x. They can be joined.
quadratic expression: an expression whose highest power of x is x², such as x² + 5x + 6.

Back to your guess

The answer is B: x² + 5x + 6: multiply every term by every term, then join the like terms.

Every term in the first bracket multiplies every term in the second. That gives four products: x², 2x, 3x and 6. The two middle terms are alike and join to make 5x. So (x + 3)(x + 2) = x² + 5x + 6.

Keep it

Copy this into your exercise book.

Multiply every term by every term. Keep each sign with its term. Join the like terms. A squared bracket is the bracket times itself, with four products. Check with a number.

Draw: A grid with x and +3 across the top, x and −5 down the side, and four products in the cells.