Lesson 11: Expanding double brackets

Introduction

By double brackets, I mean something like \(~(x+2)(x+5)~\).

To be called double brackets, the brackets have to be butted right up against each other. Don't get them mixed up with the pairs of single brackets in Lesson 9, which were separated by a \(~+~\) or \(~-~\) sign. This is a different animal!

SECTION A

The expression \(~(x+2)(x+5)~\) means \(~(x+2)\times (x+5)~\)

This is the same as saying \(~(x+2)~\) lots of \(~(x+5)~\)

Coming up with a method

Consider the expression \(~(3+2)(x+5)~\)
This means \(~(3+2)~\) lots of \(~(x+5)~\)
Which is the same as \(~3~\) lots of \(~(x+5)~\) plus another \(~2~\) lots of \(~(x+5)~\)
Which can be written as \(~3(x+5)+2(x+5)~\)
Hence \(~(x+2)(x+5)~\) can be written as \(~x(x+5)+2(x+5)~\)

Take the time to understand this and the examples will seem easy.

Example 1

Expand and simplify \(~(x+2)(x+5)~\)

Rewrite in the form \(~x(x+5)+2(x+5)~\)

Expand the single brackets ⇒ \(~x^2+5x+2x+10~\)

Collect the \(~x~\) terms ⇒ Answer: \(~x^2+7x+10~\)

Example 2

Expand and simplify \(~(x+3)(x-4)~\)

Rewrite in the form \(~x(x-4)+3(x-4)~\)

Expand the single brackets ⇒ \(~x^2-4x+3x-12~\)

Collect the \(~x~\) terms ⇒ Answer: \(~x^2-x-12~\)

Example 3

Expand and simplify \(~(x-2)(x-3)~\)

Rewrite in the form \(~x(x-3)-2(x-3)~\)

Expand the single brackets ⇒ \(~x^2-3x-2x+6~\)

Collect the \(~x~\) terms ⇒ Answer: \(~x^2-5x+6~\)

Example 4

Expand and simplify \(~(3x-2)(2x+1)~\)

Rewrite in the form \(~3x(2x+1)-2(2x+1)~\)

Expand the single brackets ⇒ \(~6x^2+3x-4x-2~\)

Collect the \(~x~\) terms ⇒ Answer: \(~6x^2-x-2~\)

A common mistake is to forget the power of 2 on the first term. You then end up collecting three \(~x~\) terms, leaving a total of only two terms in your final answer. Don't do that!

Enjoy!

Practise to master

SECTION A

Expand and simplify the following expressions.

01) \(~(x+1)(x+5)~\)

\(x(x+5)+1(x+5)\)
\(x^2+5x+1x+5\)
\(x^2+6x+5\)

02) \(~(x+4)(x+5)~\)

\(x(x+5)+4(x+5)\)
\(x^2+5x+4x+20\)
\(x^2+9x+20\)

03) \(~(x+9)(x+2)~\)

\(x(x+2)+9(x+2)\)
\(x^2+2x+9x+18\)
\(x^2+11x+18\)

04) \(~(2x+1)(x+1)~\)

\(2x(x+1)+1(x+1)\)
\(2x^2+2x+1x+1\)
\(2x^2+3x+1\)

05) \(~(x+2)(5x+4)~\)

\(x(5x+4)+2(5x+4)\)
\(5x^2+4x+10x+8\)
\(5x^2+14x+8\)

06) \(~(3x+2)(2x+3)~\)

\(3x(2x+3)+2(2x+3)\)
\(6x^2+9x+4x+6\)
\(6x^2+13x+6\)

07) \(~(x+2)(x-5)~\)

\(x(x-5)+2(x-5)\)
\(x^2-5x+2x-10\)
\(x^2-3x-10\)

08) \(~(x-4)(x+1)~\)

\(x(x+1)-4(x+1)\)
\(x^2+1x-4x-4\)
\(x^2-3x-4\)

09) \(~(x-5)(x-6)~\)

\(x(x-6)-5(x-6)\)
\(x^2-6x-5x+30\)
\(x^2-11x+30\)

10) \(~(3x+1)(x-2)~\)

\(3x(x-2)+1(x-2)\)
\(3x^2-6x+1x-2\)
\(3x^2-5x-2\)

11) \(~(x-4)(2x+1)~\)

\(x(2x+1)-4(2x+1)\)
\(2x^2+1x-8x-4\)
\(2x^2-7x-4\)

12) \(~(4x-1)(3x-4)~\)

\(4x(3x-4)-1(3x-4)\)
\(12x^2-16x-3x+4\)
\(12x^2-19x+4\)

SECTION B

This section will prepare you for questions that look slightly different, though the method is the same.

Example 1

Expand and simplify \(~(x-6)^2~\)

Anything squared is that thing times itself ⇒ \(~(x-6)(x-6)~\)

Rewrite in the form \(~x(x-6)-6(x-6)~\)

Expand the single brackets ⇒ \(~x^2-6x-6x+36~\)

Collect like terms ⇒ Answer: \(~x^2-12x+36~\)

Write the expression out without a power and the rest is ... well, no different to what you've been doing.

Example 2

Expand and simplify \(~(2x+2)(x-5)-3x~\)

Rewrite in the form \(~2x(x-5)+2(x-5)-3x~\)

Expand the single brackets ⇒ \(~2x^2-10x+2x-10-3x~\)

Collect like terms ⇒ Answer: \(~2x^2-11x-10~\)

The only difference here is the extra \(~x~\) term which needs to be carried through and collected at the end. This extra term could also be a number or an \(~x^2~\) term.

Polish these questions off and you've nailed yet another important topic!

Practise to master

SECTION B

Factorise the following expressions.

01) \(~(x+3)^2~\)

\((x+3)(x+3)\)
\(x(x+3)+3(x+3)\)
\(x^2+3x+3x+9\)
\(x^2+6x+9\)

02) \(~(x-5)(x+5)~\)

\(x(x+5)-5(x+5)\)
\(x^2+5x-5x-25\)
\(x^2-25\)
This special case is what Lesson 14 is all about!

03) \(~(x+3)(1+x)~\)

\(x(1+x)+3(1+x)\)
\(x+x^2+3+3x\)
\(x^2+4x+3\)
We write the highest powers first, remember?

04) \(~(2x+3)(x-5)+5~\)

\(2x(x-5)+3(x-5)+5\)
\(2x^2-10x+3x-15+5\)
\(2x^2-7x-10\)

05) \(~(2x+5)^2~\)

\((2x+5)(2x+5)\)
\(2x(2x+5)+5(2x+5)\)
\(4x^2+10x+10x+25\)
\(4x^2+20x+25\)

06) \(~(3x-4)(3x+1)~\)

\(3x(3x+1)-4(3x+1)\)
\(9x^2+3x-12x-4\)
\(9x^2-9x-4\)

07) \(~(2x+1)(2+3x)~\)

\(2x(2+3x)+1(2+3x)\)
\(4x+6x^2+2+3x\)
\(6x^2+7x+2\)

08) \(~(2x-7)(2x+7)+x^2~\)

\(2x(2x+7)-7(2x+7)+x^2\)
\(4x^2+14x-14x-49+x^2\)
\(5x^2-49\)

09) \(~(4x-1)^2~\)

\((4x-1)(4x-1)\)
\(4x(4x-1)-1(4x-1)\)
\(16x^2-4x-4x+1\)
\(16x^2-8x+1\)

10) \(~(5x-2)(3x+1)~\)

\(5x(3x+1)-2(3x+1)\)
\(15x^2+5x-6x-2\)
\(15x^2-x-2\)

11) \(~(3x-1)(2-x)~\)

\(3x(2-x)-1(2-x)\)
\(6x-3x^2-2+x\)
\(-3x^2+7x-2\)

12) \(~(7+2x)(4x+5)-3x~\)

\(7(4x+5)+2x(4x+5)-3x\)
\(28x+35+8x^2+10x-3x\)
\(8x^2+35x+35\)

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