Factorising & Completing the Square (DP IB Analysis & Approaches (AA)) : Revision Note

Did this video help you?

Factorising Quadratics

Why is factorising quadratics useful?

  • Factorising gives roots (zeroes or solutions) of a quadratic

  • It gives the x-intercepts when drawing the graph

How do I factorise a monic quadratic of the form x2 + bx + c?

  • A monic quadratic is a quadratic where the coefficient of the x2 term is 1

  • You might be able to spot the factors by inspection

    • Especially if c is a prime number

  • Otherwise find two numbers m and n ..

    • A sum equal to b

      • p plus q equals b

    • A product equal to c

      • p q equals c

  • Rewrite bx as mx + nx

  • Use this to factorise x2 + mx + nx + c

  • A shortcut is to write:

    • left parenthesis x plus p right parenthesis left parenthesis x plus q right parenthesis

How do I factorise a non-monic quadratic of the form ax2 + bx + c?

  • A non-monic quadratic is a quadratic where the coefficient of the x2 term is not equal to 1

  • If a, b & c have a common factor then first factorise that out to leave a quadratic with coefficients that have no common factors

  • You might be able to spot the factors by inspection

    • Especially if a and/or c are prime numbers

  • Otherwise find two numbers m and n ..

    • A sum equal to b

      • m plus n equals b

    • A product equal to ac

      • m n equals a c

  • Rewrite bx as mx + nx

  • Use this to factorise ax2 + mx + nx + c

  • A shortcut is to write:

    • fraction numerator left parenthesis a x plus m right parenthesis left parenthesis a x plus n right parenthesis over denominator a end fraction

    • Then factorise common factors from numerator to cancel with the a on the denominator

How do I use the difference of two squares to factorise a quadratic of the form a2x2 - c2?

  • The difference of two squares can be used when...

    • There is no x term

    • The constant term is a negative

  • Square root the two terms a squared x squared and c squared

  • The two factors are the sum of square roots and the difference of the square roots

  • A shortcut is to write:

    • open parentheses a x plus c close parentheses open parentheses a x minus c close parentheses

Examiner Tips and Tricks

  • You can deduce the factors of a quadratic function by using your GDC to find the solutions of a quadratic equation

    • Using your GDC, the quadratic equation  6 x squared plus x minus 2 equals 0  has solutions  x equals negative 2 over 3  and  x equals 1 half 

    • Therefore the factors would be  left parenthesis 3 x plus 2 right parenthesis  and  left parenthesis 2 x minus 1 right parenthesis

    • i.e.  6 x squared plus x minus 2 equals left parenthesis 3 x plus 2 right parenthesis left parenthesis 2 x minus 1 right parenthesis

Worked Example

Factorise fully:

a) x squared minus 7 x plus 12.

2-2-2-ib-aa-sl-factorise-a-we-solution

b) 4 x squared plus 4 x minus 15.

2-2-2-ib-aa-sl-factorise-b-we-solution

c) 18 minus 50 x squared.

2-2-2-ib-aa-sl-factorise-c-we-solution

Did this video help you?

Completing the Square

Why is completing the square for quadratics useful?

  • Completing the square gives the maximum/minimum of a quadratic function

    • This can be used to define the range of the function

  • It gives the vertex when drawing the graph

  • It can be used to solve quadratic equations

  • It can be used to derive the quadratic formula

How do I complete the square for a monic quadratic of the form x2 + bx + c?

  • Half the value of b and write stretchy left parenthesis x plus b over 2 stretchy right parenthesis squared

    • This is because stretchy left parenthesis x plus b over 2 stretchy right parenthesis squared equals x squared plus b x plus b squared over 4

  • Subtract the unwanted b squared over 4 term and add on the constant c

    • stretchy left parenthesis x plus b over 2 stretchy right parenthesis squared minus b squared over 4 plus c

How do I complete the square for a non-monic quadratic of the form ax2 + bx + c?

  • Factorise out the a from the terms involving x

    • a stretchy left parenthesis x squared plus b over a x stretchy right parenthesis plus x 

    • Leaving the c alone will avoid working with lots of fractions

  • Complete the square on the quadratic term

    • Half b over a and write stretchy left parenthesis x plus fraction numerator b over denominator 2 a end fraction stretchy right parenthesis squared

      • This is because stretchy left parenthesis x plus fraction numerator b over denominator 2 a end fraction stretchy right parenthesis squared equals x squared plus b over a x plus fraction numerator b squared over denominator 4 a squared end fraction

    • Subtract the unwanted fraction numerator b squared over denominator 4 a squared end fraction term

  • Multiply by a and add the constant c

    • a stretchy left square bracket stretchy left parenthesis x plus fraction numerator b over denominator 2 a end fraction stretchy right parenthesis squared minus fraction numerator b squared over denominator 4 a squared end fraction stretchy right square bracket plus c

    • a stretchy left parenthesis x plus fraction numerator b over denominator 2 a end fraction stretchy right parenthesis squared minus fraction numerator b squared over denominator 4 a end fraction plus c

Examiner Tips and Tricks

  • Some questions may not use the phrase "completing the square" so ensure you can recognise a quadratic expression or equation written in this form

    • a left parenthesis x minus h right parenthesis squared plus k space left parenthesis equals 0 right parenthesis

Worked Example

Complete the square:

a) x squared minus 8 x plus 3.

2-2-2-ib-aa-sl-complete-square-a-we-solution

b) 3 x squared plus 12 x minus 5.

2-2-2-ib-aa-sl-complete-square-b-we-solution
👀 You've read 1 of your 5 free revision notes this week
An illustration of students holding their exam resultsUnlock more revision notes. It's free!

By signing up you agree to our Terms and Privacy Policy.

Already have an account? Log in

Did this page help you?

Dan Finlay

Author: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

Download notes on Factorising & Completing the Square