Expanding double brackets

Two brackets multiplied together are opened by multiplying everything in the one bracket with everything in the other, and then simplifying. It's the same idea as multiplying a number into a bracket, just done twice over. The hurdle is keeping track of all four multiplications and the signs that come with them.

When do I use this?

Whenever you have something like $(x + 1)(x - 2)$ and you want it written out as ordinary terms. The place this shows up most is with quadratic functions. Sometimes we're not interested in having a factorised quadratic and would rather rewrite it to the standard form, that is

$$f(x) = ax^2 + bx + c$$

We can do that super easily just by simplifying, in other words by multiplying our brackets out. The opposite direction, from the standard form to the brackets, is in factorising quadratics.

The procedure

Let's take an example. We have the quadratic

$$f(x) = 2(x + 1)(x - 2)$$

1. Start by multiplying the two brackets together. We do that by multiplying everything in the one bracket with everything in the other bracket. In the book there are four arrows: from the $x$ in the first bracket to the $x$ and the $-2$ in the second, and from the $1$ in the first bracket to the $x$ and the $-2$ in the second. That gives us

$$(x + 1)(x - 2) = x^2 - 2x + x - 2$$

2. Simplify what you got. The $-2x$ and the $+x$ are like terms, so we can just simplify this to

$$x^2 - x - 2$$

3. Then deal with the number in front. Now that we've multiplied the brackets together, we just have to multiply by 2, because there's a 2 in front of the brackets:

$$2 \times (x^2 - x - 2)$$

We do the same as before. The 2 has to be multiplied with everything in the bracket, which gives us

$$2x^2 - 2x - 4$$

Now we're done working it out, and so our quadratic turns out to be

$$f(x) = 2x^2 - 2x - 4$$

Remember that the factorised form is exactly the same. It's exactly the same graph, we've just rewritten it.

Squared brackets

A bracket squared is just the bracket multiplied by itself, so it's opened the same way. $(a + b)^2$ means $(a + b)(a + b)$. Everything in the one with everything in the other gives $a \times a$, $a \times b$, $b \times a$ and $b \times b$, and the two middle products are like terms:

$$(a + b)^2 = a^2 + b^2 + 2ab$$

With a minus inside it's the same four multiplications, but the two middle ones come out negative:

$$(a - b)^2 = a^2 + b^2 - 2ab$$

These two are worth recognising, because they turn up again later on. If you forget them, you can always multiply the brackets out like any other pair.

Worked examples

The example from the book. Expand $2(x + 1)(x - 2)$.

Brackets first: $(x + 1)(x - 2) = x^2 - 2x + x - 2 = x^2 - x - 2$. Then the 2 into everything: $2x^2 - 2x - 4$.

$$2(x + 1)(x - 2) = 2x^2 - 2x - 4$$

Two brackets on their own. Expand $(x - 3)(x + 5)$.

Everything in the one with everything in the other: $x \times x = x^2$, $x \times 5 = 5x$, $-3 \times x = -3x$ and $-3 \times 5 = -15$.

$$(x - 3)(x + 5) = x^2 + 5x - 3x - 15 = x^2 + 2x - 15$$

Notice that the minus in front of the 3 comes along into both of its products. The sign belongs to the term.

A squared bracket. Expand $(x + 4)^2$.

That's $(x + 4)(x + 4)$, so $x^2 + 4x + 4x + 16$, which simplifies to $x^2 + 8x + 16$. We can check this is right, because we know $(a + b)^2 = a^2 + b^2 + 2ab$, and with $a = x$ and $b = 4$ that's $x^2 + 16 + 8x$. Same thing.

Common mistakes

• "Thinking the expanded form is a different function." It isn't. $2(x + 1)(x - 2)$ and $2x^2 - 2x - 4$ are exactly the same graph, just written two ways.
• "Forgetting the number in front." The 2 in $2(x + 1)(x - 2)$ has to be multiplied with everything in the bracket you end up with, not just the first term.
• "Mixing up the like terms." After the four multiplications you'll have two middle terms, like $-2x$ and $+x$. Those are like terms and get collected. $x^2$ and $x$ are not, they're apples and bananas, and stay apart.
• "Dropping a minus." In $(x - 3)(x + 5)$ the $-3$ takes its sign into both of its products, $-3x$ and $-15$.

Related

The single-bracket version of this move, including why a minus in front flips the signs, is in expanding a single bracket. The tidying-up at the end is collecting like terms. Why quadratics get written with brackets in the first place is in quadratic functions and factorising quadratics.