Algebraic Expressions and Formulae

Key Concepts

• Algebra uses letters and numbers to represent values and relationships.

  • Letters such as $x$, $y$, and $a$ are called variables.
  • A variable can stand for an unknown value or a value that can change.

• An algebraic expression is a mathematical phrase made up of numbers, variables, and operations such as $+$, $-$, $\times$, and $\div$.

  • Examples: $3x+5$, $2a^2-7a$, $\frac{x+1}{3}$

• A term is one part of an expression, separated by $+$ or $-$.

  • In $4x^2-3x+6$, the terms are $4x^2$, $-3x$, and $6$.

• A coefficient is the numerical part of a term.

  • In $7y$, the coefficient is $7$.
  • In $-2a^2$, the coefficient is $-2$.

• A constant is a fixed number without a variable.

  • In $x^2+4x-9$, $-9$ is the constant.

• Like terms are terms with the same variable part.

  • $3x$ and $-5x$ are like terms.
  • $2a^2$ and $7a^2$ are like terms.
  • $4x$ and $4x^2$ are not like terms.

1. Expansion of Products of Algebraic Expressions

• Expansion means removing brackets by multiplying each term inside the bracket.

• Use the distributive law:

  $$a(b+c)=ab+ac$$

Expanding a single bracket

• Multiply the term outside the bracket by every term inside.

• Example:

  $$3(x+4)=3x+12$$

Expanding two brackets

• Multiply every term in the first bracket by every term in the second bracket.

• Example:

  $$(x+2)(x+5)=x^2+5x+2x+10=x^2+7x+10$$

• After expanding, always collect like terms.

2. Standard Algebraic Identities

An identity is an equation that is true for all values of the variables.

You must know these standard identities:

Square of a sum

Square of a difference

Product of a sum and a difference

Important notes

• $(a+b)^2$ does not mean $a^2+b^2$
• The middle term comes from multiplying the two terms twice:
  • $ab+ab=2ab$

3. Factorisation of Algebraic Expressions

• Factorisation is the reverse of expansion.
• It means writing an expression as a product of factors.

Common factor

• Find the highest common factor of all the terms and take it out.

• Example:

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

Factorising by grouping

• Sometimes terms can be grouped to form common factors.

• Example:

  $$ax+ay+bx+by=a(x+y)+b(x+y)=(a+b)(x+y)$$

Using identities

• Recognise patterns:
  • $a^2+2ab+b^2=(a+b)^2$
  • $a^2-2ab+b^2=(a-b)^2$
  • $a^2-b^2=(a+b)(a-b)$

4. Factorising Quadratic Expressions $ax^2+bx+c$

• A quadratic expression has highest power $2$.

• General form:

  $$ax^2+bx+c$$

  where $a\neq 0$

When $a=1$

• Factorise $x^2+bx+c$ by finding two numbers:
  • whose product is $c$
  • whose sum is $b$

Example:

because $3\times4=12$ and $3+4=7$

When $a\neq1$

• Find two numbers:
  • whose product is $ac$
  • whose sum is $b$
• Then split the middle term and factorise by grouping.

Example:

• $a\times c=2\times3=6$
• Need two numbers that multiply to $6$ and add to $7$: $6$ and $1$

So,

5. Changing the Subject of a Formula

• A formula is an equation that shows a relationship between quantities.

  • Example:

    $$A=\frac{1}{2}bh$$

• The subject of a formula is the variable that is written alone on one side.

  • In $A=\frac{1}{2}bh$, the subject is $A$

• Changing the subject means rearranging the formula so that a different variable is the subject.

Steps

1. Identify the variable to make the subject.
2. Remove addition or subtraction first.
3. Remove multiplication or division next.
4. Keep the equation balanced by doing the same operation to both sides.

Example:

Make $h$ the subject.

• Multiply both sides by $2$:

  $$2A=bh$$

• Divide both sides by $b$:

  $$h=\frac{2A}{b}$$

6. Operations with Algebraic Fractions

An algebraic fraction is a fraction that contains variables.

Example:

Simplifying algebraic fractions

• Factorise the numerator and denominator first.
• Cancel only common factors, not terms.

Example:

Multiplication

• Multiply numerators together and denominators together.
• Factorise first if possible.

Example:

Division

• Change division into multiplication by the reciprocal.
• Reciprocal means flipping the fraction.

Example:

Addition and subtraction

• Find a common denominator first.
• Then combine the numerators.

Example:

Example with different denominators:

Common denominator is $3x$:

So,

Restriction

• The denominator of a fraction cannot be zero.
• If $x-3$ is in the denominator, then $x\neq3$

Important Definitions

• Variable: a letter that represents an unknown value or a value that can change.
• Algebraic expression: a combination of numbers, variables, and operations.
• Term: a part of an algebraic expression separated by $+$ or $-$.
• Coefficient: the numerical factor of a term.
• Constant: a number without a variable.
• Like terms: terms with the same variable part.
• Expansion: removing brackets by multiplication.
• Distributive law: the rule $a(b+c)=ab+ac$.
• Identity: an equation that is true for all values of the variables.
• Factorisation: writing an expression as a product of factors.
• Common factor: a factor shared by two or more terms.
• Quadratic expression: an expression with highest power $2$, usually written as $ax^2+bx+c$.
• Formula: an equation showing the relationship between quantities.
• Subject of a formula: the variable that is alone on one side of the equation.
• Algebraic fraction: a fraction involving variables.
• Reciprocal: the multiplicative inverse of a number or fraction; for $\frac{a}{b}$, the reciprocal is $\frac{b}{a}$.

Worked Examples

Example 1: Expand and simplify $(2x+3)(x-4)$

Step 1: Multiply each term in the first bracket by each term in the second bracket

Step 2: Simplify each product

Step 3: Collect like terms

Answer:

Example 2: Change the subject of $v=u+at$ to $t$

We want $t$ to be the subject.

Step 1: Remove $u$ from the right side

Subtract $u$ from both sides:

Step 2: Remove $a$

Divide both sides by $a$:

Answer:

Example 3: Factorise $2x^2+5x-3$

Step 1: Identify $a$, $b$, and $c$

Step 2: Find $ac$

Step 3: Find two numbers with product $-6$ and sum $5$

These are $6$ and $-1$

Step 4: Split the middle term

Step 5: Factorise by grouping

Step 6: Take out common bracket

Answer:

Common Mistakes to Avoid

• Expanding $(a+b)^2$ as $a^2+b^2$

  • Correct form is $a^2+2ab+b^2$

• Forgetting to multiply every term in a bracket.

  • In $(x+2)(x+3)$, you must get 4 products.

• Sign errors when multiplying negatives.

  • Example: $(-3)(+2)=-6$, $(-3)(-2)=+6$

• Collecting unlike terms.

  • $3x+2x^2$ cannot be simplified further.

• Cancelling terms instead of factors in algebraic fractions.

  • $\frac{x+3}{x}$ cannot be simplified by cancelling the $x$

• Forgetting to factorise before simplifying fractions.

  • Example: $\frac{x^2-4}{x-2}$ must first become $\frac{(x-2)(x+2)}{x-2}$

• Rearranging a formula incorrectly by moving terms without using inverse operations.

  • Always do the same operation to both sides.

• Leaving the final answer unsimplified.

  • Always collect like terms or cancel common factors where possible.

• Forgetting restrictions for algebraic fractions.

  • If the denominator is $x-5$, then $x\neq5$

Exam Tips

• When expanding brackets, write out all products clearly before simplifying.
• Use identities only when the expression matches the pattern exactly.
• In factorisation, always check your answer by expanding the factors again.
• For $x^2+bx+c$, look for two numbers with:
  • product = constant term
  • sum = coefficient of $x$
• For $ax^2+bx+c$, first calculate $ac$.
• When changing the subject of a formula:
  • state or show each operation on both sides
  • keep the target variable positive if possible
• In algebraic fractions:
  • factorise first
  • state common denominator when adding or subtracting
  • cancel only common factors
• Watch signs carefully, especially when the constant term is negative.
• If your factorised answer has brackets, make sure each bracket is as simple as possible.

Quick Summary

• An algebraic expression contains variables, numbers, and operations.
• Expansion means removing brackets by multiplication.
• Use the distributive law: $a(b+c)=ab+ac$
• $(a+b)^2=a^2+2ab+b^2$
• $(a-b)^2=a^2-2ab+b^2$
• $(a+b)(a-b)=a^2-b^2$
• Factorisation is the reverse of expansion.
• To factorise, first look for a common factor.
• For $x^2+bx+c$, find two numbers whose product is $c$ and sum is $b$.
• For $ax^2+bx+c$, find two numbers whose product is $ac$ and sum is $b$, then group.
• To change the subject of a formula, use inverse operations on both sides.
• For algebraic fractions, factorise first and cancel only common factors, not terms.