Introduction: Two Approaches to Solving Equations

In IB Mathematics AA, you will encounter equations that range from straightforward to surprisingly complex. Two core strategies exist for tackling them:

Analytical methods — using algebraic manipulation to find exact solutions
Graphical methods — using graphs (by hand or with technology) to find approximate or exact solutions visually

Knowing when to use each approach — and being able to move between them — is a key skill tested in both Paper 1 (no calculator) and Paper 2 (calculator allowed).

Note

Paper 1 will almost always require analytical methods, since no calculator is available. Paper 2 often allows or even expects graphical/technology-based approaches for complex equations.

Analytical Methods: Algebraic Techniques

Analytical methods involve rearranging and manipulating an equation using algebraic rules to isolate the unknown and find exact solutions.

Common techniques include:

Factorisation — rewriting an expression as a product of factors
Substitution — replacing a complex expression with a simpler variable to reveal a recognisable structure (e.g. a hidden quadratic)
Using logarithms — to solve exponential equations by taking $ln$ or $lo g$ of both sides. Note: logarithms are only defined for strictly positive arguments, so always check that both sides are positive before applying $ln$ or $lo g$
Using the quadratic formula — $x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$ for equations of the form $a x^{2} + b x + c = 0$

Exact Solution

A solution expressed as a precise value, often involving surds, fractions, or logarithms, rather than a decimal approximation. For example, $x = ln 4$ rather than $x \approx 1.386$ .

The discriminant and number of solutions

Before solving a quadratic $a x^{2} + b x + c = 0$ , the discriminant $Δ = b^{2} - 4 a c$ tells you how many real solutions exist:

Discriminant	Number of real solutions
$Δ > 0$	Two distinct real solutions
$Δ = 0$	One repeated real solution
$Δ < 0$	No real solutions

This is useful both analytically (before solving) and graphically (to anticipate how many times the parabola crosses the $x$ -axis).

Exam Tip

Always look for a hidden quadratic structure when you see expressions like $e^{2 x}$ , $sin^{2} x$ , or $(f (x))^{2}$ alongside $f (x)$ . A substitution $u = f (x)$ often transforms the equation into a familiar quadratic.

Free preview

SL 2.10—Solving equations graphically and analytically

Introduction: Two Approaches to Solving Equations

Analytical Methods: Algebraic Techniques

9 more sections in this topic