Unit 11 Section 2 : Simplifying Expressions

In this section we look at how to simplify expressions, in particular, how to remove brackets from both formulae and equations.

Collecting like terms

Examples

a + a + a = 3a

a + b + a = 2a + b

2y + 8y = 10y

x + x^2 + x^2 = x + 2x^2

Only like terms can be collected

Example 1

Simplify the following expressions,

(a)4a + 2b + 3a + 6b = 7a + 8b
(b)3x - 4y + 2x - y = 5x - 5y
(c)x^2 + 4x + 2x^2 - x = 3x^2 + 3x
(d)4a^2 + a + 2a^2 - 3a = 6a^2 - 2a

Expanding Brackets

Every term in each bracket must be multiplied by every other item.

x(4x + 2) = x × 4x + x × 2
= 4x^2 + 2x
(x + 1)(x + 4) = x × x + x × 4 + 1 × x + 1 × 4
= x^2 + 4x + x + 4
= x^2 + 5x + 4
Alternatively, you can expand brackets using the 'box' method, as shown opposite.

(x + 1)(x + 4) = x^2 + 1x + 4x + 4 = x^2 + 5x + 4

Example 2

Expand each of the following:

(a)2(x + 3) 2(x + 3) = 2 × x + 2 × 3 = 2x + 6
(b)4(2x - 6) 4(2x - 6) = 4 × 2x - 4 × 6 = 8x - 24
(c)x(x + 2) x(x + 2) = x × x + x × 2 =x^2 + 2x
(d)2x(3x - 2) 2x(3x - 2) = 2x × 3x - 2x × 2 = 6x^2 - 4x

Example 3

Expand,

(a)

(x + 6)(x + 3)

(x + 6)(x + 3) = x × x + x × 3 + 6 × x + 6 × 3
= x^2 + 3x + 6x + 18
= x^2 + 9x + 18

or alternatively, using the box method,

(x + 6)(x + 3) = x^2 + 6x + 3x + 18
= x^2 + 9x + 18
(b)

(x + 4)(2x - 5)

(x + 4)(2x - 5) = x × 2x - x × 5 + 4 × 2x - 4 × 5
= 2x^2 - 5x + 8x - 20
= 2x^2 + 3x - 20

Again, using the box method,

(x + 4)(2x - 5) = 2x^2 + 8x - 5x - 20
= 2x^2 + 3x - 20

Exercises

Note: To type indeces on this page use ^ sign. e.g.  n2
Question 1

Simplify each of the following by collecting like terms:

(a)4a + b + 2a
(b)4b + 2c + 6b + 3c
(c)4a + 5b - a + 2b
(d)14p + 11q - 8p + 3q
(e)6x - 4y + 8x + 9y
(f)11x + 8y + 3z - 2y + 4z
(g)16x - 8y - 3x - 4y
(h)11y + 12z - 10y + 4z + 2y
Question 2

Simplify each of the following:

(a)3x + 3x^2 + 4x - x^2
(b)4y^2 + 4y - 2y^2 + 3y
(c)a^2 + a + 3a^2 - 2a
(d)6x^2 + 12x - 9x^2 + 3x
Question 3

Expand each of the following expressions by multiplying out the brackets:

(a)3(x + 6)
(b)4(x + 2)
(c)3(x - 1)
(d)4(2x + 5)
(e)6(3x - 5)
(f)7(2x - 5)
(g)6(4 - 2x)
(h)8(3 - 5x)
(i)9(5x + 10)
Question 4

Simplify each of the following expressions:

(a)2(x + 3) + 4(x + 4)
(b)5(x - 6) + 2(x + 3)
(c)4(6 - x) + 7(2x + 1)
(d)11(x - 3) + 4(7x + 3)
(e)8(x - 6) + 4(7 - x)
(f)3(4 - 5x) + 6(3x - 2)
Question 5

Expand each of the following expressions by multiplying out the brackets:

(a)x(x + 3)
(b)x(6x + 1)
(c)x(3x - 2)
(d)2x(4 - x)
(e)6x(2x + 4)
(f)5x(3x - 7)
(g)11x(x - 3)
(h)14x(2 + 3x)
(i)6x(4 - 2x)
Question 6

Expand each of the following expressions by multiplying out the brackets:

(a)(x + 4)(x + 3)
(b)(x + 2)(x + 4)
(c)(x + 1)(x + 5)
(d)(x + 6)(x - 1)
(e)(x - 4)(x + 2)
(f)(x - 3)(x + 2)
(g)(x - 4)(x - 5)
(h)(x - 3)(x - 2)
(i)(x - 7)(x - 9)
Question 7

Simplify each of the following expressions:

(a)(x + 2)(x + 4) + (x + 1)(x + 2)
(b)(x + 3)(x + 7) + (x - 1)(x + 5)
(c)(x + 6)(x + 2) - (x - 2)(x + 3)
(d)(x - 4)(x - 8) - (x - 1)(x - 9)
Question 8

Expand each expression:

(a)(2x + 1)(3x + 2)
(b)(4x - 7)(2x + 1)
(c)(3x + 5)(2x - 8)
(d)(4x + 5)(3x - 8)
(c)(8x + 2)(3x - 3)
(d)(6x - 5)(3x - 7)
Question 9

Simplify:

(a)(3x + 2)(5x + 9) + (4x - 2)(3x - 5)
(b)(4x + 6)(5x + 1) - (2x + 3)(3x + 1)
(c)(6x - 5)(x + 1) - (2x + 7)(3x - 5)
Question 10

Expand:

(a)(x + 1)^2
(b)(x - 2)^2
(c)(x + 3)^2
(d)(x + 5)^2
(e)(x - 7)^2
(f)(x - 8)^2
(g)(x + 10)^2
(h)(x - 12)^2
(i)(x + 4)^2
(j)(2x + 3)^2
(k)(4x - 7)^2
(l)(3x + 2)^2
(m)(4x + 1)^2
(n)(5x - 2)^2
(o)(6x - 4)^2
Question 11

Expand:

(a)(x + 1)(x - 1)
(b)(x + 3)(x - 3)
(c)(x + 7)(x - 7)
(d)(x + 9)(x - 9)
(e)(x + 12)(x - 12)
(f)(2x + 1)(2x - 1)
(g)(3x + 2)(3x - 2)
(h)(4x + 7)(4x - 7)
Question 12

Expand:

(a)(x + 1)^3
(b)(2x + 1)^3
(c)(x - 5)^3
Question 13

Here are some algebra cards:

Note: To use division sign ÷ write / instead.
(a)

One of the cards will always give the same answer as .

Which card is it?
(b)

One of the cards will always give the same answer as .

Which card is it?
(c)

Two of the cards will always give the same answer as .

Which cards are they?
and
Question 14

(a)
(i)

The diagram shows a rectangle 18 cm long and 14 cm wide. It has been split into four smaller rectangles, A, B, C and D.
Write down the area of each of the small rectangles.

One has been done for you.

Area of Rectangle A = cm².

Area of Rectangle B = cm².

Area of Rectangle C = 40 cm².

Area of Rectangle D = cm².

(ii)

What is the area of the whole rectangle?

cm²
(iii)

What is 18 × 14 ?

(b)
(i)

The diagram shows a rectangle (n + 3) cm long and (n + 2) cm wide. It has been split into four smaller rectangles.
Write down a number or an expression for the area of each small rectangle.

One has been done for you.

Area of Rectangle E = cm².

Area of Rectangle F = 3n cm².

Area of Rectangle G = cm².

Area of Rectangle H = cm².

(ii)

What is (n + 3)(n + 2) multiplied out?

Question 15

Multiply out and simplify these expressions:

(a)3(x - 2) - 2(4 - 3x)
(b)(x + 2)(x + 3)
(c)(x + 4)(x - 1)
(c)(x - 2)^2
Question 16

A number grid is inside a large triangle. The small triangles are numbered consecutively.
The diagram shows the first 4 rows.

(a)

An expression for the last number in row n is n2.

Write an expression for the last but one number in row n.

(b)

An expression for the first number in row n is n^2 - 2n + 2.
Calculate the value of the first number in row 10.

(c)

Make a copy of the table and complete it by writing an expression:

first number in row nn^2 - 2n + 2
second number in row n
(d)

Make a copy of the table and complete it by writing an expression:

centre number in row nn^2 - n + 1
centre number in row (n + 1)^2 - (n + 1) + 1
(e)

Multiply out and simplify the expression (n + 1)^2 - (n + 1) + 1.
Show your working.

(n + 1)^2 - (n + 1) + 1 =
=