Expanding Double Brackets

To expand two brackets, such as (x+3)(x+5), means to multiply them out so that the answer has no brackets. Every term in the first bracket multiplies every term in the second bracket.

Two terms times two terms gives four terms. Usually two of them are like terms, so you collect them. This is what expand and simplify means:

(x+3)(x+5)=x^2+5x+3x+15=x^2+8x+15

In the exam you are usually asked to expand and simplify two brackets, such as (x+4)(x-7), for 2 marks. You can get 1 mark for three of the four terms right, with the right signs, so write all four terms down before you collect them. Harder questions have a number in front of the letter, such as (2x+3)(4x-1), or give the sides of a rectangle or square as brackets and ask about its area.

You can picture (x+2)(x+3) as the area of a rectangle that is x+2 wide and x+3 tall. Split each side into its two parts, and the rectangle splits into four smaller rectangles.

The four areas are x\times x=x^2, x\times2=2x, 3\times x=3x and 3\times2=6. Add them up and collect the two x terms:

(x+2)(x+3)=x^2+2x+3x+6=x^2+5x+6

Try it: set the slider to x=4. The rectangle is 6 by 7, so its area is 42. The four parts give the same total: 16+8+12+6=42

In the exam you don't need to draw the rectangle. A grid does the same job: write the terms of one bracket along the top and the terms of the other down the side, then multiply to fill each box.

Without a grid, take the first term of the first bracket and multiply it by both terms in the second bracket. Then do the same with the second term of the first bracket. That gives the four terms.

Each term keeps the sign in front of it. In (x-3)(x+5) the terms are x and -3, then x and 5:

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

Use the sign rules. A positive times a negative is negative, so -3\times5=-15. A negative times a negative is positive, so -3\times(-5)=15

(x+3)^2 means (x+3)(x+3). Write the bracket out twice, then expand as before:

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

It is not x^2+9, because squaring a bracket gives a middle term too. Check with x=1: (1+3)^2=16 and 1+6+9=16, but 1^2+9=10

When two brackets are the same apart from the sign, the middle terms cancel. For example, (x+6)(x-6)=x^2-6x+6x-36=x^2-36. This is called the difference of two squares.

Some brackets have a number in front of the letter, such as (3x+2). When you multiply two terms such as 3x and 5x, multiply the numbers, then multiply the letters.

3x\times5x=15x^2

The 15 comes from 3\times5, and the x^2 comes from x\times x. The answer is 15x^2, not 15x or 8x

For example, in (2x+1)(x-3) the four terms are 2x\times x=2x^2, 2x\times(-3)=-6x, 1\times x=x and 1\times(-3)=-3. So:

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

The area of a rectangle is length × width. If a rectangle is (x+6) cm long and (x+1) cm wide, its area is (x+6)(x+1). Expand and simplify to write the area without brackets:

(x+6)(x+1)=x^2+x+6x+6=x^2+7x+6

A square has four equal sides, so a square with sides of (x+2) cm has area (x+2)^2, which is x^2+4x+4

Some questions tell you the area as a number and ask you to show that an equation is true. Expand the brackets, write the result equal to the area, then rearrange it as you would an equation. Write down every step, because the answer is given and the marks are for the working.

• Writing only two terms. (x+2)(x+6) is not x^2+12: multiply every term by every term to get four terms, then collect.
• Losing the squared term. x\times x=x^2, not 2x, and 3x\times2x=6x^2, not 6x or 5x
• Getting a sign wrong. A negative times a negative is positive, so -5\times(-2)=10, but -5\times2=-10
• Collecting the middle terms wrongly. -2x+6x=4x, not 8x or -8x
• Squaring each term on its own. Write the bracket out twice: (x+4)^2 is x^2+8x+16, not x^2+16
• Simplifying too far. x^2+2x-15 is finished, because x^2, 2x and -15 are not like terms.