What is the fundamental difference between factorising $x^2 - 16$ and $x^2 + 16$?

$x^2 - 16$ is a difference of two squares and factorises to $(x-4)(x+4)$. $x^2 + 16$ is a sum of squares and cannot be factorised into linear factors using real numbers.

When factorising $ax^2 + bx + c$, how does the process change if $a = 1$ versus $a \neq 1$?

If $a=1$, you find factors of $c$ that sum to $b$ and place them directly in brackets. If $a \neq 1$, you find factors of $ac$ that sum to $b$, split the middle term, and factorise by grouping.

How does the goal of factorising differ from the goal of expanding?

Expanding uses the distributive property to turn a product of factors into a sum of terms. Factorising reverses this to turn a sum of terms into a product of factors, which is useful for solving equations.

What is the most common error when factorising an expression like $2x^2 - 8$?

The most common error is forgetting to extract the HCF ($2$) first. If you extract the $2$, you get $2(x^2 - 4)$, which can then be further factorised into $2(x-2)(x+2)$.

Why is it incorrect to factorise $x^2 - 5x - 6$ as $(x-2)(x-3)$?

While $-2$ and $-3$ add to $-5$, their product is $+6$, not $-6$. To get a product of $-6$ and a sum of $-5$, the correct factors are $-6$ and $+1$, resulting in $(x-6)(x+1)$.

What happens if you forget to check for a negative HCF when the leading term is negative (e.g., $-x^2 + 5x - 6$)?

Failing to factor out $-1$ first makes finding the correct factor pairs much more difficult. Extracting $-1$ gives $-(x^2 - 5x + 6)$, which is much easier to factorise as $-(x-2)(x-3)$.

Define a 'Monic Quadratic' in the context of factorisation.

A monic quadratic is a quadratic expression where the leading coefficient ($a$) is equal to $1$, such as $x^2 + bx + c$. These are generally the simplest quadratics to factorise using the sum and product method.

What is the 'Difference of Two Squares' formula?

The formula is $a^2 - b^2 = (a-b)(a+b)$. It states that the difference between two squared terms is equal to the product of the difference and the sum of their square roots.

What does the 'Highest Common Factor' (HCF) represent in algebra?

The HCF is the largest coefficient and the highest power of any variables that are present in every single term of the expression. It is the first thing removed during factorisation.

Why must the two numbers $p$ and $q$ multiply to $c$ and add to $b$ in $x^2 + bx + c$?

This comes from expanding $(x+p)(x+q) = x^2 + qx + px + pq = x^2 + (p+q)x + pq$. By comparing this to $x^2 + bx + c$, we see that $b$ must equal $p+q$ and $c$ must equal $pq$.

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2. Algebra & Functions

Factorising Quadratics

Summary

Factorising quadratics is the algebraic process of breaking down a quadratic expression into a product of simpler linear factors. It serves as the inverse operation to expanding brackets and is a foundational skill for solving quadratic equations, simplifying rational expressions, and analyzing parabolic functions.

1. Definition & Core Concepts

A flowchart showing the logical steps for factorising: starting with the expression, extracting the HCF, and then choosing between the Difference of Two Squares for binomials or the Trinomial method for three-term expressions.

2. Underlying Principles

3. Methods & Techniques

4. Key Distinctions

5. Exam Strategy & Tips

The HCF Priority: Always check for a common factor across all terms before applying other methods. Failing to do so often makes the numbers much harder to work with and results in an incomplete factorisation.
Verification by Expansion: You can always check if your factorisation is correct by mentally or physically expanding your answer using the FOIL method ( $First, Outside, Inside, Last$ ). If you don't get the original expression back, an error was made.
Sign Consistency: Pay close attention to the signs of $b$ and $c$ . If $c$ is positive, both factors must have the same sign as $b$ . If $c$ is negative, the factors must have opposite signs, with the larger factor taking the sign of $b$ .

6. Common Pitfalls & Misconceptions

Factorising Quadratics

Summary

1. Definition & Core Concepts

A quadratic expression is a polynomial of degree 2, typically written in the standard form $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants and $a \neq 0$ .

Factorising is the process of rewriting this sum or difference of terms as a product of two or more factors, such as $(px + q)(rx + s)$ .

The Highest Common Factor (HCF) is the largest term that divides exactly into every term of the expression and should always be extracted first to simplify the factorisation process.

2. Underlying Principles

Factorising relies on the Distributive Law in reverse; while expansion distributes a term across a sum, factorising identifies the shared components to group them back into brackets.

For a monic trinomial $x^2 + bx + c$ , the logic is based on the expansion $(x+p)(x+q) = x^2 + (p+q)x + pq$ . Therefore, we seek two numbers $p$ and $q$ that satisfy $p+q=b$ and $pq=c$ .

The Null Factor Law is the primary reason we factorise; it states that if $A \times B = 0$ , then either $A=0$ or $B=0$ , allowing us to solve quadratic equations once they are in factored form.

3. Methods & Techniques

Difference of Two Squares (DOTS)

This method applies to binomials in the form $a^2 - b^2$ , which always factorise to $(a-b)(a+b)$ .
It requires both terms to be perfect squares separated by a subtraction sign; it cannot be applied to a sum of squares like $a^2 + b^2$ .

Simple Trinomials ( $a=1$ )

To factorise $x^2 + bx + c$ , identify two integers that multiply to give the constant term $c$ and add to give the coefficient $b$ .
Once found, the expression is written directly as $(x + ext{first number})(x + ext{second number})$ .

Non-Monic Trinomials ( $a \neq 1$ )

Use the splitting the middle term method by finding two numbers that multiply to $ac$ and add to $b$ .
Rewrite the middle term $bx$ using these two numbers, then factorise by grouping the first two terms and the last two terms separately.

4. Key Distinctions

Choosing the correct method depends entirely on the structure and number of terms in the expression.

Feature	Difference of Two Squares	Perfect Square Trinomial	General Trinomial
Structure	$a^2 - b^2$	$a^2 \pm 2ab + b^2$	$ax^2 + bx + c$
Terms	2 terms	3 terms	3 terms
Result	$(a-b)(a+b)$	$(a \pm b)^2$	$(px+q)(rx+s)$
Requirement	Must be subtraction	Middle term is $2 \times \sqrt{a} \times \sqrt{c}$	Sum/Product relationship

5. Exam Strategy & Tips

The HCF Priority: Always check for a common factor across all terms before applying other methods. Failing to do so often makes the numbers much harder to work with and results in an incomplete factorisation.
Verification by Expansion: You can always check if your factorisation is correct by mentally or physically expanding your answer using the FOIL method ( $First, Outside, Inside, Last$ ). If you don't get the original expression back, an error was made.
Sign Consistency: Pay close attention to the signs of $b$ and $c$ . If $c$ is positive, both factors must have the same sign as $b$ . If $c$ is negative, the factors must have opposite signs, with the larger factor taking the sign of $b$ .

6. Common Pitfalls & Misconceptions

Sum of Squares Error: Students often try to factorise $x^2 + 9$ as $(x+3)(x+3)$ or $(x-3)(x+3)$ . However, $x^2 + 9$ is prime (cannot be factorised) because the middle term $0x$ cannot be produced by these combinations.
Incomplete Factorisation: A common mistake is stopping after extracting an HCF or only partially factorising a complex expression. Always check if the terms inside the resulting brackets can be factorised further (e.g., a remaining $x^2 - 4$ should become $(x-2)(x+2)$ ).
Incorrect Factor Pairs: When $c$ is large, students often pick the first factor pair they find that multiplies to $c$ without checking if they add to $b$ . Systematically listing factor pairs helps avoid this oversight.

A quadratic expression is a polynomial of degree 2, typically written in the standard form $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants and $a \neq 0$ .

Factorising is the process of rewriting this sum or difference of terms as a product of two or more factors, such as $(px + q)(rx + s)$ .

The Highest Common Factor (HCF) is the largest term that divides exactly into every term of the expression and should always be extracted first to simplify the factorisation process.

Factorising relies on the Distributive Law in reverse; while expansion distributes a term across a sum, factorising identifies the shared components to group them back into brackets.

For a monic trinomial $x^2 + bx + c$ , the logic is based on the expansion $(x+p)(x+q) = x^2 + (p+q)x + pq$ . Therefore, we seek two numbers $p$ and $q$ that satisfy $p+q=b$ and $pq=c$ .

The Null Factor Law is the primary reason we factorise; it states that if $A \times B = 0$ , then either $A=0$ or $B=0$ , allowing us to solve quadratic equations once they are in factored form.

Difference of Two Squares (DOTS)

This method applies to binomials in the form $a^2 - b^2$ , which always factorise to $(a-b)(a+b)$ .
It requires both terms to be perfect squares separated by a subtraction sign; it cannot be applied to a sum of squares like $a^2 + b^2$ .

Simple Trinomials ( $a=1$ )

To factorise $x^2 + bx + c$ , identify two integers that multiply to give the constant term $c$ and add to give the coefficient $b$ .
Once found, the expression is written directly as $(x + ext{first number})(x + ext{second number})$ .

Non-Monic Trinomials ( $a \neq 1$ )

Use the splitting the middle term method by finding two numbers that multiply to $ac$ and add to $b$ .
Rewrite the middle term $bx$ using these two numbers, then factorise by grouping the first two terms and the last two terms separately.

Choosing the correct method depends entirely on the structure and number of terms in the expression.

Feature	Difference of Two Squares	Perfect Square Trinomial	General Trinomial
Structure	$a^2 - b^2$	$a^2 \pm 2ab + b^2$	$ax^2 + bx + c$
Terms	2 terms	3 terms	3 terms
Result	$(a-b)(a+b)$	$(a \pm b)^2$	$(px+q)(rx+s)$
Requirement	Must be subtraction	Middle term is $2 \times \sqrt{a} \times \sqrt{c}$	Sum/Product relationship

Sum of Squares Error: Students often try to factorise $x^2 + 9$ as $(x+3)(x+3)$ or $(x-3)(x+3)$ . However, $x^2 + 9$ is prime (cannot be factorised) because the middle term $0x$ cannot be produced by these combinations.
Incomplete Factorisation: A common mistake is stopping after extracting an HCF or only partially factorising a complex expression. Always check if the terms inside the resulting brackets can be factorised further (e.g., a remaining $x^2 - 4$ should become $(x-2)(x+2)$ ).
Incorrect Factor Pairs: When $c$ is large, students often pick the first factor pair they find that multiplies to $c$ without checking if they add to $b$ . Systematically listing factor pairs helps avoid this oversight.

Factorising Quadratics

Summary

1. Definition & Core Concepts

2. Underlying Principles

3. Methods & Techniques

4. Key Distinctions

5. Exam Strategy & Tips

6. Common Pitfalls & Misconceptions

Factorising Quadratics

Summary

1. Definition & Core Concepts

2. Underlying Principles

3. Methods & Techniques

Difference of Two Squares (DOTS)

Simple Trinomials (a=1a=1a=1)

Non-Monic Trinomials (a≠1a \neq 1a=1)

4. Key Distinctions

5. Exam Strategy & Tips

6. Common Pitfalls & Misconceptions

Difference of Two Squares (DOTS)

Simple Trinomials (a=1a=1a=1)

Non-Monic Trinomials (a≠1a \neq 1a=1)

Simple Trinomials ( $a=1$ )

Non-Monic Trinomials ( $a \neq 1$ )

Simple Trinomials ( $a=1$ )

Non-Monic Trinomials ( $a \neq 1$ )