DP IB Maths: AA SL

Revision Notes

2.2.2 Factorising & Completing the Square

Test Yourself

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

Exam Tip

  • 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

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

Exam Tip

  • 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

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Dan

Author: Dan

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.