Solving Quadratics | DP IB Maths: AA HL Revision Notes 2021

Solving Quadratic Equations

How do I decide the best method to solve a quadratic equation?

A quadratic equation is of the form $a x^{2} + b x + c = 0$
If it is a calculator paper then use your GDC to solve the quadratic
If it is a non-calculator paper then...
- you can always use the quadratic formula
- you can factorise if it can be factorised with integers
- you can always complete the square

How do I solve a quadratic equation by the quadratic formula?

If necessary rewrite in the form $a x^{2} + b x + c = 0$
Clearly identify the values of a, b & c
Substitute the values into the formula
- $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$
  - This is given in the formula booklet
Simplify the solutions as much as possible

How do I solve a quadratic equation by factorising?

Factorise to rewrite the quadratic equation in the form $a (x - p) (x - q) = 0$
Set each factor to zero and solve
- $x - p = 0 \Rightarrow x = p$
- $x - q = 0 \Rightarrow x = q$

How do I solve a quadratic equation by completing the square?

Complete the square to rewrite the quadratic equation in the form $a {(x - h)}^{2} + k = 0$
Get the squared term by itself
- ${(x - h)}^{2} = - \frac{k}{a}$
If $(- \frac{k}{a})$ is negative then there will be no solutions
If $(- \frac{k}{a})$ is positive then there will be two values for $x - h$
- $x - h = \pm \sqrt{- \frac{k}{a}}$
Solve for x
- $x = h \pm \sqrt{- \frac{k}{a}}$

Examiner Tip

When using the quadratic formula with awkward values or fractions you may find it easier to deal with the " $b^{2} - 4 a c$ " (discriminant) first
- This can help avoid numerical and negative errors, improving accuracy

Worked example

Solve the equations:

4 x^{2} + 4 x - 15 = 0

2-2-3-ib-aa-sl-quadratic-equations-a-we-solution

3 x^{2} + 12 x - 5 = 0

2-2-3-ib-aa-sl-quadratic-equations-b-we-solution

7 - 3 x - 5 x^{2} = 0

2-2-3-ib-aa-sl-quadratic-equations-c-we-solution

Solving Quadratics (DP IB Maths: AA HL)

Revision Note

How do I decide the best method to solve a quadratic equation?

How do I solve a quadratic equation by the quadratic formula?

How do I solve a quadratic equation by factorising?

How do I solve a quadratic equation by completing the square?

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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

2.7.4 Roots of Polynomials

2.8 Inequalities

2.8.1 Solving Inequalities Graphically