Factorising Quadratics

• Quadratic Expression: A quadratic expression is a polynomial of degree two, typically written in the general form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants and $a \neq 0$. The highest power of the variable (usually $x$) is 2.

• Factorising: Factorising a quadratic expression means rewriting it as a product of two or more simpler expressions, usually linear factors. For example, $x^2 + 5x + 6$ can be factorised into $(x+2)(x+3)$. This process is the reverse of expanding (multiplying out) brackets.

• Linear Factors: When a quadratic is factorised, it typically results in two linear factors, each of the form $(dx+e)$. These factors represent the expressions that, when multiplied together, yield the original quadratic expression.

• Distributive Property: The foundation of factorising lies in the distributive property of multiplication over addition, which states that $A(B+C) = AB + AC$. When factorising, we are essentially reversing this process, identifying a common factor $A$ to pull out from $AB+AC$.

• Reverse of Expansion: Factorising is the inverse operation of expanding brackets. For instance, expanding $(x+p)(x+q)$ gives $x^2 + qx + px + pq = x^2 + (p+q)x + pq$. Therefore, to factorise $x^2 + bx + c$, we seek two numbers $p$ and $q$ such that their sum is $b$ and their product is $c$.

• Zero Product Property (Implicit): While factorising itself doesn't solve equations, the utility of factorised form for solving relies on the Zero Product Property. This property states that if the product of two or more factors is zero, then at least one of the factors must be zero. This is why factorising is a crucial step before setting each factor to zero to find roots.

• Common Factor Method: If a quadratic expression has no constant term (i.e., $c=0$), or if all terms share a common numerical or variable factor, the first step is to factor out the greatest common factor. For example, $3x^2 + 12x$ can be factorised by taking out $3x$, resulting in $3x(x+4)$.

• Inspection Method (for $a=1$): For quadratics of the form $x^2 + bx + c$, the inspection method involves finding two numbers, let's call them $p$ and $q$, that satisfy two conditions: their product $p \times q$ must equal the constant term $c$, and their sum $p + q$ must equal the coefficient of the $x$ term, $b$. Once these numbers are found, the quadratic factorises directly into $(x+p)(x+q)$.

• Systematic Search: To find $p$ and $q$ for the inspection method, systematically list pairs of factors of $c$ and check their sums. Pay close attention to the signs of $b$ and $c$; if $c$ is positive, $p$ and $q$ have the same sign (both positive if $b$ is positive, both negative if $b$ is negative); if $c$ is negative, $p$ and $q$ have opposite signs.

• Grouping Method (Splitting the Middle Term): This is a reliable method for factorising quadratics where the coefficient of $x^2$, $a$, is not 1. The process begins by finding two numbers, $p$ and $q$, whose product is $a \times c$ and whose sum is $b$. These numbers are then used to rewrite the middle term, $bx$, as $px + qx$.

• Step-by-Step Grouping: Once the middle term is split, the quadratic $ax^2 + px + qx + c$ is grouped into two pairs: $(ax^2 + px)$ and $(qx + c)$. A common factor is then extracted from each pair, leading to an expression like $X(Y) + Z(Y)$. Finally, the common bracket $(Y)$ is factored out, resulting in $(X+Z)(Y)$.

• Trial and Error (Inspection for $a \neq 1$): While not always systematic, experienced mathematicians can sometimes factorise harder quadratics by inspection. This involves considering factors of $a$ and $c$ and arranging them in binomials $(dx+e)(fx+g)$ such that the inner and outer products sum to $bx$. This method requires significant practice and is less reliable than grouping for complex cases.

• Recognition: A difference of two squares is a quadratic expression of the form $a^2 - b^2$. It is characterized by two perfect square terms separated by a subtraction sign. Examples include $x^2 - 4$, $9y^2 - 25$, or even $(x+1)^2 - (y-2)^2$.

• Factorisation Rule: The general rule for factorising a difference of two squares is:

$a^2 - b^2 = (a+b)(a-b)$ This rule is derived directly from expanding the right-hand side: $(a+b)(a-b) = a^2 - ab + ab - b^2 = a^2 - b^2$.

• Application: To apply this rule, identify the 'first term squared' ($a^2$) and the 'second term squared' ($b^2$). Then, take the square root of each term to find $a$ and $b$, and substitute them into the $(a+b)(a-b)$ form. This method is often quicker and simpler than other factorisation techniques when applicable.

• Prioritizing Methods: Always begin by checking for a common factor among all terms, regardless of the quadratic's form. Factoring out a common factor simplifies the remaining expression and makes subsequent steps easier. For example, $2x^2 + 10x + 12$ should first be factored as $2(x^2 + 5x + 6)$.

• Choosing Between Inspection and Grouping: If the coefficient of $x^2$ is 1 (i.e., $x^2 + bx + c$), the inspection method is generally the most efficient. If $a \neq 1$ (i.e., $ax^2 + bx + c$), the grouping method is a systematic and reliable approach, though inspection can be used with practice.

• Recognizing Special Forms: Always be on the lookout for the difference of two squares pattern ($a^2 - b^2$). This pattern allows for direct factorisation into $(a+b)(a-b)$, which is significantly faster than other methods. It's also important to recognize perfect square trinomials, though the document doesn't explicitly cover them, they are related to factorising.

• Verification by Expansion: After factorising any quadratic, a crucial step is to expand your answer (multiply out the brackets) to ensure it matches the original expression. This simple check can catch most errors related to signs or incorrect factors, providing immediate feedback on accuracy.

• Sign Errors: A very common mistake is incorrect signs when finding $p$ and $q$. Carefully consider the signs of $b$ and $c$ in $x^2+bx+c$ or $ac$ and $b$ in $ax^2+bx+c$. For example, if $c$ is positive, $p$ and $q$ must have the same sign; if $c$ is negative, they must have opposite signs.

• Missing Common Factors: Students often forget to look for or factor out a common factor at the beginning. This can make the quadratic harder to factorise or lead to an incomplete factorisation. Always ensure the final factors are in their simplest form, meaning no further common factors can be extracted from within the brackets.

• Not Recognizing Difference of Two Squares: Failing to identify a difference of two squares can lead to using more complex methods unnecessarily or even incorrectly. Train yourself to spot this pattern quickly, as it offers the most direct path to factorisation.