Factorising & Completing the Square | DP IB Maths: AA HL Revision Notes 2021

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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
  - $p + q = b$
- A product equal to c
  - $p q = c$
Rewrite bx as mx + nx
Use this to factorise x² + mx + nx + c
A shortcut is to write:
- $(x + p) (x + q)$

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
  - $m + n = b$
- A product equal to ac
  - $m n = a c$
Rewrite bx as mx + nx
Use this to factorise ax² + mx + nx + c
A shortcut is to write:
- $\frac{(a x + m) (a x + n)}{a}$
- 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 $a^{2} x^{2}$ and $c^{2}$
The two factors are the sum of square roots and the difference of the square roots
A shortcut is to write:
- $(a x + c) (a x - c)$

Examiner 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 $6 x^{2} + x - 2 = 0$ has solutions $x = - \frac{2}{3}$ and $x = \frac{1}{2}$
- Therefore the factors would be $(3 x + 2)$ and $(2 x - 1)$
- i.e. $6 x^{2} + x - 2 = (3 x + 2) (2 x - 1)$

Worked example

Factorise fully:

x^{2} - 7 x + 12

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

4 x^{2} + 4 x - 15

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

18 - 50 x^{2}

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

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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 ${(x + \frac{b}{2})}^{2}$
- This is because ${(x + \frac{b}{2})}^{2} = x^{2} + b x + \frac{b^{2}}{4}$
Subtract the unwanted $\frac{b^{2}}{4}$ term and add on the constant c
- ${(x + \frac{b}{2})}^{2} - \frac{b^{2}}{4} + c$

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
- $a (x^{2} + \frac{b}{a} x) + x$
- Leaving the c alone will avoid working with lots of fractions
Complete the square on the quadratic term
- Half $\frac{b}{a}$ and write ${(x + \frac{b}{2 a})}^{2}$
  - This is because ${(x + \frac{b}{2 a})}^{2} = x^{2} + \frac{b}{a} x + \frac{b^{2}}{4 a^{2}}$
- Subtract the unwanted $\frac{b^{2}}{4 a^{2}}$ term
Multiply by a and add the constant c
- $a [{(x + \frac{b}{2 a})}^{2} - \frac{b^{2}}{4 a^{2}}] + c$
- $a {(x + \frac{b}{2 a})}^{2} - \frac{b^{2}}{4 a} + c$

Examiner 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
- $a {(x - h)}^{2} + k (= 0)$

Worked example

Complete the square:

x^{2} - 8 x + 3

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

3 x^{2} + 12 x - 5

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

Factorising & Completing the Square (DP IB Maths: AA HL)

Revision Note

Why is factorising quadratics useful?

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

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

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

Why is completing the square for quadratics useful?

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

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

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1. Number & Algebra

1.1 Number & Algebra Toolkit

1.1.1 Standard Form

1.1.2 Laws of Indices

1.1.3 Partial Fractions

1.2 Exponentials & Logs

1.2.1 Introduction to Logarithms

1.2.2 Laws of Logarithms

1.2.3 Solving Exponential Equations

1.3 Sequences & Series

1.3.1 Language of Sequences & Series

1.3.2 Arithmetic Sequences & Series

1.3.3 Geometric Sequences & Series

1.3.4 Applications of Sequences & Series

1.3.5 Compound Interest & Depreciation

1.4 Simple Proof & Reasoning

1.4.1 Proof

1.5 Further Proof & Reasoning

1.5.1 Proof by Induction

1.5.2 Proof by Contradiction

1.6 Binomial Theorem

1.6.1 Binomial Theorem

1.6.2 Extension of The Binomial Theorem

1.7 Permutations & Combinations

1.7.1 Counting Principles

1.7.2 Permutations & Combinations

1.8 Complex Numbers

1.8.1 Intro to Complex Numbers

1.8.2 Modulus & Argument

1.8.3 Introduction to Argand Diagrams

1.9 Further Complex Numbers

1.9.1 Geometry of Complex Numbers

1.9.2 Forms of Complex Numbers

1.9.3 Complex Roots of Polynomials

1.9.4 De Moivre's Theorem

1.9.5 Roots of Complex Numbers

1.10 Systems of Linear Equations

1.10.1 Systems of Linear Equations

1.10.2 Algebraic Solutions

2. Functions

2.1 Linear Functions & Graphs

2.1.1 Equations of a Straight Line

2.2 Quadratic Functions & Graphs

2.2.1 Quadratic Functions

2.2.2 Factorising & Completing the Square

2.2.3 Solving Quadratics

2.2.4 Quadratic Inequalities

2.2.5 Discriminants

2.3 Functions Toolkit

2.3.1 Language of Functions

2.3.2 Composite & Inverse Functions

2.3.3 Symmetry of Functions

2.3.4 Graphing Functions

2.4 Other Functions & Graphs

2.4.1 Exponential & Logarithmic Functions

2.4.2 Solving Equations

2.4.3 Modelling with Functions

2.5 Reciprocal & Rational Functions

2.5.1 Reciprocal & Rational Functions

2.6 Transformations of Graphs

2.6.1 Translations of Graphs

2.6.2 Reflections of Graphs

2.6.3 Stretches Graphs

2.6.4 Composite Transformations of Graphs

2.7 Polynomial Functions

2.7.1 Factor & Remainder Theorem

2.7.2 Polynomial Division

2.7.3 Polynomial Functions

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

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

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

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

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