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Factorising Quadratic Expressions

Factorizing Quadratic Expressions involves expressing them as a product of two or more Algebraic Expressions or Polynomial Expressions. The goal is to rewrite a quadratic expression in a form that makes its properties or solutions more apparent, or simply it to extracting common factor and quadratic expressions. This process is important in various mathematical contexts, including algebraic manipulation and solving equations.

In this chapter, we will be discussing the below-mentioned topics in detail:

  • Factorisation as the reverse process of Expansion
  • Factorisation using a Multiplication Frame
  • Factorisation using a Factorisation Table
  • Factorisation involving extracting common factor and quadratic expressions

Factorisation of Algebraic Expressions

Types Of Factorisation \(\begin{align} \xrightarrow [] {\quad \text{ Expansion }\quad } \end{align}\)
Extract Common Factor \(\begin{align*} 2x(x-1) &= \end{align*}\) \(\begin{align*} 2x^2-2x \end{align*}\)
Quadratic Factorisation \(\begin{align} (x-2)(x-1) &= \end{align}\) \(\begin{align} x^2-3x+2 \end{align}\)
  \(\xleftarrow{\quad \text{ Factorisation }\quad}\)

Factorisation is the opposite of expansion. 

Let’s understand this with the help of some examples:

Question 1: 

Factorise \(3x^2-27x\) completely.

Solution: 

The common factor is \(3x\).

\(3x^2-27=3x(x-9)\)

 

Question 2: 

Factorise \(x^2+5x+6\) completely.

Solution:

Method (a): Factorisation using Multiplication Frame

\(x^2+5x+6\)

Factorisation using Multiplication Frame

As we know, \(x\) times \(x\) will give us \(x^2\), and plot both the \(x\) in Row \(\text{I}\) Column \(\text{II}\) and Row \(\text{II}\) Column \(\text{I}\), respectively. 

Also, either 3 times 2 is 6 or 1 times 6 is 6, so using trial and error, put 3 in Row \(\text{III}\) Column \(\text{I}\) and put 2 in Row \(\text{I}\) Column \(\text{III}\).

Multiply Row \(\text{I}\) Column \(\text{III}\), i.e. 2, by Row \(\text{II}\) Column \(\text{I}\), i.e. \(x\) and put \(2x\) in Row \(\text{II}\) Column \(\text{III}\).

Multiply Row \(\text{III}\) Column \(\text{I}\), i.e. 3, by Row \(\text{I}\) Column \(\text{II}\), i.e. \(x\) and plot \(3x\) in Row \(\text{III}\) Column \(\text{II}\).

Now, \(+2x\) and \(+3x\) give us \(+5x\)

However, if we use 1 times 6 instead, see below.

factorisation using multiplication frame 2

Now, \(+6x\) and \(+x\) do not give us \(+5x\), and it gives us \(+7x\).

Hence, the answer would be \((x+2)(x+3)\)

 

Method (b): Factorisation using Factorisation Table

Factorisation using Factorisation Table

\(\therefore \quad x^2+5x+6=(x+3)(x+2)\)

 

Question 3: 

Factorise \(x^2+6x+5\) completely.

Solution: 

Using a factorisation table,

Factorisation using Factorisation Table 2

\(\therefore \quad x^2+6x+5=(x+1)(x+5)\)

 

Question 4:

Factorise \(3x^2-18x+15\) completely.

Solution:

Firstly, we extract out a common factor 3.

\(3x^2-18x+15=3(x^2-6x+5)\)

Factorisation using Factorisation Table 3

\(\begin{align*} \therefore \quad 3x^2-18x+15 &= 3(x^2-6x+5)\\[2ex] &=3(x-1)(x-5) \end{align*}\)

 


 

Continue Learning
Algebraic Fractions Direct & Inverse Proportion
Congruence And Similarity Factorising Quadratic Expressions
Further Expansion And Factorisation Quadratic Equations And Graphs
Simultaneous Equation

 

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