Expressions, Formulae, and Equations | Year 8 Mathematics (Cambridge) PDF Download

Constructing Expressions

What is an Expression?

An expression is a combination of numbers, variables, and mathematical operations (such as addition, subtraction, multiplication, and division). For example, 3x + 5 and 2y − 7 are expressions.

How to Construct Expressions

To construct an expression, follow these steps:

• Identify the variables: These are the symbols that represent unknown values (e.g., x or y).
• Choose the constants: These are the fixed numbers that do not change (e.g., 3, 5, 7).
• Combine with operations: Use addition, subtraction, multiplication, or division to combine variables and constants.

Example:

• Problem: John has x apples, and he buys 5 more. How can we represent the total number of apples?
• Expression: x + 5

Using Expressions and Formulae

What is a Formula?

A formula is a mathematical rule expressed using symbols. It shows the relationship between different variables. For example, the formula for the area of a rectangle is A = l × w, where A is the area, l is the length, and w is the width.

How to Use Expressions in Formulae

To use expressions in formulae, substitute the values of variables into the expression and then perform the calculations.

Example:
Problem: The formula for the perimeter of a rectangle is P = 2l + 2w. If the length l = 4 and the width w = 3, find the perimeter.
Solution: Substitute l = 4 and w = 3 into the formula:
P = 2(4) + 2(3)
Calculate: P = 8 + 6 = 14

Expanding Brackets

What is Expanding Brackets?

Expanding brackets means to multiply out the terms inside the brackets. For example, to expand 3(x + 4), multiply 3 by both x and 4.

How to Expand Brackets

• Distribute the term outside the brackets: Multiply it by each term inside the brackets.
• Simplify the expression: Combine like terms if necessary.

Example:
Problem: Expand 2(x + 5).
Solution: Distribute 2 to both x and 5:
2 ⋅ x + 2⋅5
Simplify: 2x + 10

Factorising

What is Factorising?

Factorising is the reverse process of expanding. It involves writing an expression as a product of its factors.

How to Factorise

• Find the common factor: Identify the greatest common factor of the terms.
• Rewrite the expression: Express the terms as a product of the common factor and the remaining terms inside brackets.

Example:
Problem: Factorise 4x + 8.
Solution: Identify the common factor (4):
4(x + 2)

Factorising Quadratic Expressions

To factorise quadratic expressions like ax² + bx + c, look for two numbers that multiply to ac and add to b.

Example:
Problem: Factorise x² + 5x + 6.
Solution: Find two numbers that multiply to 6 and add to 5 (2 and 3): (x + 2)(x + 3)