What is the difference between a monic and a non-monic quadratic expression?

A monic quadratic has a leading coefficient ($a$) of 1, such as $x^2 + bx + c$. A non-monic quadratic has a leading coefficient other than 1, such as $3x^2 + bx + c$, which usually requires more complex factorisation methods like grouping.

How does the factorisation of $x^2 - y^2$ differ from $x^2 + y^2$?

$x^2 - y^2$ is a 'Difference of Two Squares' and factorises to $(x - y)(x + y)$. In contrast, $x^2 + y^2$ is a 'Sum of Two Squares' and cannot be factorised into linear factors using real numbers.

When should you use the 'Inspection' method versus the 'Grouping' method?

Inspection is best for monic quadratics ($a=1$) where the factors of $c$ that add to $b$ are easily spotted. Grouping is necessary for non-monic quadratics ($a \neq 1$) to systematically split the middle term and factorise in pairs.

What is a common mistake when factorising quadratics with a negative constant term ($c$)?

Students often forget that a negative constant means the two factors must have different signs (one positive, one negative). This requires finding two numbers whose *difference* is the middle coefficient $b$, rather than their sum.

Why is it an error to ignore a common numerical factor before factorising a quadratic?

Ignoring a common factor makes the coefficients larger and more difficult to work with, and the resulting factors will not be 'fully factorised'. For example, $2x^2 + 10x + 12$ should be simplified to $2(x^2 + 5x + 6)$ first.

What happens if you incorrectly identify the factors of $ac$ in the grouping method?

If the chosen factors do not multiply to $ac$ AND add to $b$, the pairwise factorisation step will fail because a common binomial bracket will not appear. This indicates that either the factors are wrong or the quadratic is not factorable.

Define the 'Discriminant' in the context of factorisation.

The discriminant is the value $b^2 - 4ac$ from the quadratic formula. If this value is a perfect square, the quadratic expression can be factorised into brackets with rational coefficients.

What is the 'Difference of Two Squares' identity?

It is the algebraic rule $a^2 - b^2 = (a - b)(a + b)$. It applies whenever an expression consists of one perfect square subtracted from another, with no linear ($x$) term present.

What does it mean to 'factorise fully'?

To factorise fully means to extract all possible common factors (numerical and variable) and then break down the remaining expression into its simplest possible linear factors.

Why must the two numbers in the grouping method multiply to $ac$ rather than just $c$?

Because the leading coefficient $a$ is distributed across the factors during expansion. Multiplying to $ac$ accounts for the interaction between the quadratic coefficient and the constant term in the final expanded form.

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2. Algebra & Functions

Quadratics Factorising Methods

Summary

Factorising quadratics is the process of reversing expansion to express a second-degree polynomial as a product of linear factors. This fundamental algebraic skill relies on identifying relationships between coefficients and constants, utilizing techniques ranging from simple inspection to structured grouping and the 'Difference of Two Squares' identity.

1. Definition & Core Concepts

2. Underlying Principles

The logic of factorisation is based on the distributive property in reverse. When we expand $(x + p)(x + q)$ , we get $x^2 + (p+q)x + pq$ , which shows that the middle coefficient is the sum of two numbers and the constant is their product.
For non-monic quadratics, the relationship is more complex: the two numbers used to split the middle term must multiply to the product of the leading coefficient and the constant ( $ac$ ) and add to the middle coefficient ( $b$ ).
The Zero Product Property is the primary motivation for factorising; it states that if the product of two factors is zero, at least one of the factors must be zero, allowing us to find the roots of the equation.

3. Methods for Monic Quadratics ($a=1$)

A 2x2 grid showing the placement of quadratic terms for factorisation. The x-squared term and constant c are in diagonal boxes, while the split linear terms occupy the other two boxes. Row and column highest common factors (HCF) form the final brackets.

4. Methods for Non-Monic Quadratics ($a \neq 1$)

5. Key Distinctions

6. Exam Strategy & Tips