Grade 12 Mathematics Study Guide

The Grade 12 Mathematics year is the most demanding of your school career, and it builds directly on skills you first met in Grade 10 and 11. Think of a learner in Polokwane preparing a small business plan for an Economics project: before she can calculate whether her R5 000 startup loan will grow under interest, she must be fluent in exponents, factorisation and solving equations. This chapter is your toolkit. It does not introduce new sequences or calculus; instead it sharpens the algebra you will lean on in every chapter that follows, so that the new content does not stumble on old gaps.

1.1 The number system and laws of exponents

South African CAPS works inside the real numbers, written \( \mathbb{R} \), which split into rational numbers (any \( \frac{a}{b} \) with \( b \neq 0 \)) and irrational numbers such as \( \sqrt{2} \) and \( \pi \). You must control the laws of exponents without hesitation, because they reappear in finance, functions and calculus.

The core laws, for any non-zero base, are \( a^m \cdot a^n = a^{m+n} \), \( \dfrac{a^m}{a^n} = a^{m-n} \), \( (a^m)^n = a^{mn} \), \( a^0 = 1 \), \( a^{-n} = \dfrac{1}{a^n} \) and the rational exponent \( a^{\frac{m}{n}} = \sqrt[n]{a^m} \).

1.2 Surds and rationalising

A surd is an irrational root such as \( \sqrt{5} \). You simplify surds using \( \sqrt{ab} = \sqrt{a}\,\sqrt{b} \), and you rationalise a denominator by multiplying top and bottom by a suitable surd so that no root remains below the line.

1.3 Factorisation: the master skill

Almost every algebra question reduces to factorising. Keep these patterns ready:

• Common factor: \( 6x^2 - 9x = 3x(2x - 3) \)
• Difference of two squares: \( a^2 - b^2 = (a-b)(a+b) \)
• Trinomials: \( x^2 + 5x + 6 = (x+2)(x+3) \)
• Sum and difference of two cubes: \( a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2) \)

The cube patterns are new and important for Grade 12, since they support the factor theorem you will use to sketch cubic graphs later in the year.

1.4 Solving equations and inequalities

You must solve linear, quadratic and simple exponential equations confidently. A quadratic equation \( ax^2 + bx + c = 0 \) is solved by factorising, or by the quadratic formula \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \] The expression \( b^2 - 4ac \) is the discriminant, written \( \Delta \); it tells you the nature of the roots before you even solve.

For an inequality such as \( x^2 - x - 6 < 0 \), factorise to \( (x-3)(x+2) < 0 \) and use a number line: the product is negative between the roots, so \( -2 < x < 3 \). Remember to flip the inequality sign whenever you multiply or divide both sides by a negative number.

1.5 Planning your Grade 12 year

Mathematics in Grade 12 is examined in two papers. Paper 1 covers algebra, sequences, functions, finance, calculus and probability; Paper 2 covers analytical geometry, trigonometry, Euclidean geometry and statistics. Work daily, redo past papers from the Department of Basic Education, and never skip the toolkit drills above. A learner who can factorise and apply exponent laws in seconds frees her mind for the genuinely new reasoning each later chapter demands.