Factorising and solving quadratic equations

At this level students will factorise monic and non-monic quadratic expressions and use factorisation techniques to solve quadratic equations. Factorisation is the opposite to expanding. It is the process of finding factors for the expression. A factorisation can always be checked by re-expanding the brackets to see if it equals the original expression.

A monic quadratic expression is an expression where the coefficient of $x^2$ is 1. For example: $x^2+6x+8$ is a monic quadratic expression.  A non-monic quadratic expression is an expression where the coefficient of $x^2$ is not 1. For example: $3x^2+5x-1$ and $\frac{x^2}{2}-2x+4$ are examples of non-monic quadratic expressions.

There are several different methods for factorisation depending on the expression.

Factorising expressions with two terms (binomials).

1. Taking out a common factor. Remove the highest common factor outside the bracket.

$2x^2-8x=2x\left(x-4\right)$  

2. Difference of two squares (DOTS). If two squared terms are being separated by a subtraction symbol, then DOTS is possible. The rule for DOTS is $a^2-b^2=\left(a-b\right)\left(a+b\right)$ . This factorisation rule is explored further in a teaching activity.

$x^2-16y^2=\left(x-4y\right)\left(x+4y\right)$  

Factorising expressions with three terms (trinomials).

1. The monic expression $x^2+bx+c$ can be factorised by looking for factors of $c$ that also add to equal $b$ .

$x^2+7x+10$  

The sum needs to be 7

The product needs to be 10

$x^2+7x+10=\left(x+5\right)\left(x+2\right)$  

2. The non-monic expression $a{x}^2+bx+c$ where $a\ne 1$ can be factorised by looking for factors of $ac$ that also add to $b$ . Once these factors have been identified the expression is written as four terms and is factorised by grouping.

$4x^2+4x-3$  

$ac=4\times -3=-12$       and     $b=4$  

$4x^2+6x-2x-3$       Use the factors to split $4x$ into $6x$ and $-2x$  

$(4x^2+6x)\left(-2x-3\right)$       Group the expression into 2 groups of 2

$2x\left(2x+3\right)-1\left(2x+3\right)$       Factorise each group by removing a common factor. This process should result in a common factor (identical bracket as highlighted). If there is not a common factor, then the factors used to split $4x$ were not correct and the process needs to be restarted.

$\therefore =\left(2x+3\right)\left(2x-1\right)$  

Factorising expressions with four terms

Group the terms into pairs. Then factorise each pair by removing a common factor. This should result in a common factor (as shown by the highlighted brackets below). This common factor can then be taken out the front and the remaining terms make up the other factor.

$2x^2+10x+3xy+15y$  

$(2x^2+10x)\left(+3xy+15y\right)$  

$2x\left(x+5\right)+3y\left(x+5\right)$  

$=\left(x+5\right)\left(2x+3y\right)$  

Support student understanding of factorising by encouraging students to re-expand to check if they have factorised correctly. It is also helpful for students to make notes of the different factorising techniques listed above and then select one depending on how many terms the expression has.  Factorising can appear a difficult task if the purpose of factorising is unknown. Support students to understand that factorising is used to solve quadratic equations and to find the x-intercepts for graphing. When students are unaware of the purpose of factorising it can seem as though it is a lot of effort to just write an expression in a different form.

Solving quadratic equations

Solving quadratic equations has been introduced in VCMNA342 and students will extend on this knowledge by creating quadratic equations to fit a given context and then solve the equation using the Null Factor Law. The Null Factor Law states that if a product of two numbers equals 0 then one or both numbers must be 0. The Null Factor Law can be used in quadratics by making the expression equal to zero, factorising and then calculating what the unknown is so both factors equal 0.

Having students substitute in the x values and check that the equation equals zero supports their understanding of why there are two values that satisfy the equation.

Quadratic equations are also used for modelling real situations. Students should be given opportunities to answer modelling questions and see how the factorising and solving skills they have learnt can be applied. Modelling questions can often be difficult for students as they may be unsure what the question is asking and what value of the quadratic equation needs to be calculated. Support student understanding by focusing on key words in quadratic modelling. This idea is explored further in a teaching activity.