Summary Revision Notes for Grade 12 NSC Matric Mathematics

Summary

Algebraic expressions

Algebraic expressions are mathematical statements that contain variables, numbers, and operations. Working with these expressions requires understanding how to combine and simplify them correctly.

Combining like terms and simplifying expressions

Like terms are terms that have exactly the same variable parts. You can only add or subtract like terms together.

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When working with algebraic expressions, always identify like terms first before attempting to combine them. This prevents common errors in simplification.

When adding or subtracting algebraic expressions, combine only the like terms
Example: $3x + 5x = 8x$ (both terms contain x)
You cannot combine $3x + 5y$ since x and y are different variables

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Worked Example: Combining Like Terms

Simplify: $4x^2 + 3x - 2x^2 + 7x - 5$

Step 1: Group like terms together

$(4x^2 - 2x^2) + (3x + 7x) - 5$

Step 2: Combine like terms

$2x^2 + 10x - 5$

Multiplication and division operations

When multiplying algebraic expressions, use the distributive law:

$\boxed{(a + b)(c + d) = ac + ad + bc + bd}$

For division:

Divide the coefficients (numbers) separately
Subtract the exponents when the bases are the same
Example: $6x^3 ÷ 2x^2 = 3x^1 = 3x$

Factorising techniques

Factorising means writing an expression as a product of its factors. This is the reverse of expanding brackets.

Common factor method

Find the highest common factor of all terms and factor it out.

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Worked Example: Common Factor Method

Factorise: $6x + 9$

Step 1: Find the highest common factor

The factors of 6x are: 1, 2, 3, 6, x, 2x, 3x, 6x
The factors of 9 are: 1, 3, 9
The highest common factor is 3

Step 2: Factor out the common factor

$6x + 9 = 3(2x + 3)$

Difference of squares

Difference of squares follows the pattern:

$\boxed{a^2 - b^2 = (a - b)(a + b)}$

This method works when you have two perfect squares being subtracted.

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The difference of squares formula only works for subtraction. You cannot use this method for $a^2 + b^2$ as this does not factorise over the real numbers.

Factorising trinomials

Trinomials are expressions with three terms, typically in the form $x^2 + bx + c$ .

Look for two numbers that multiply to give the constant term and add to give the coefficient of x
Check your answer by expanding the factored form

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Worked Example: Factorising Trinomials

Factorise: $x^2 + 5x + 6$

Step 1: Find two numbers that multiply to 6 and add to 5

$2 × 3 = 6$ and $2 + 3 = 5$ ✓

Step 2: Write in factored form

$x^2 + 5x + 6 = (x + 2)(x + 3)$

Step 3: Check by expanding

$(x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6$ ✓

Notes on factorising trinomials of the form ax² + bx + c

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When the coefficient of $x^2$ is not 1, the process becomes more complex. Find two numbers that:

Multiply to give $ac$ (coefficient of $x^2$ times the constant term)
Add to give $b$ (coefficient of $x$ )

This is often the most challenging type of factorisation and may require the quadratic formula if factoring is difficult.

Quadratic equations

Quadratic equations are equations where the highest power of the variable is 2. They can be written in the standard form $ax^2 + bx + c = 0$ .

Solving methods

There are three main methods to solve quadratic equations:

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1. Factorising Method Write the equation in factored form and set each factor equal to zero. This is the quickest method when the expression factors easily.

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2. Completing the Square Rearrange to make a perfect square trinomial. This method always works but can be time-consuming.

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3. Quadratic Formula Use the formula $x = \frac{-b ± \sqrt{b^2-4ac}}{2a}$ when other methods are difficult. This is the most reliable method for complex equations.

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Exam Strategy: Always check which method is most appropriate for the given equation. Factorising is fastest when possible, but the quadratic formula works for any quadratic equation.

Quadratic inequalities

Quadratic inequalities involve expressions with quadratic terms and inequality signs (>, <, ≥, ≤).

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Steps to Solve Quadratic Inequalities:

Solve the related quadratic equation first to find the boundary points
Sketch the parabola to visualise the solution
Use the graph to identify where the expression is positive or negative
Write the solution in interval notation or inequality form

Simultaneous equations

Simultaneous equations are systems of equations that must be solved together to find values that satisfy all equations.

Solution methods

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Substitution Method Solve one equation for one variable, then substitute this expression into the other equation. This method works well when one equation is easily solved for a variable.

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Elimination Method Add or subtract equations to eliminate one variable, then solve for the remaining variable. This method is efficient when coefficients can be easily manipulated.

These methods work for both linear-linear and linear-quadratic systems.

The nature of roots

The discriminant tells us about the nature of solutions to quadratic equations without actually solving them.

For a quadratic equation $ax^2 + bx + c = 0$ , the discriminant is:

$\boxed{\Delta = b^2 - 4ac}$

Discriminant conditions

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Nature of Roots Based on Discriminant:

$\Delta > 0$ : The equation has two distinct real roots
$\Delta = 0$ : The equation has one real repeated root
$\Delta < 0$ : The equation has two complex roots (no real solutions)

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Exam Strategy: Always calculate the discriminant first when asked about the nature of roots. This gives you immediate information about the solution type before attempting to solve the equation.

Remember!

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Key Points to Remember:

Like terms only: You can only add or subtract terms with identical variable parts
Distributive law: $(a + b)(c + d) = ac + ad + bc + bd$ - essential for expanding brackets
Difference of squares: $a^2 - b^2 = (a - b)(a + b)$ - a quick factorising shortcut
Discriminant formula: $\Delta = b^2 - 4ac$ determines the nature of quadratic equation roots
Solution methods: Choose the most efficient method (factorising, completing the square, or quadratic formula) based on the equation structure

Summary (Grade 12 NSC Matric Mathematics): Revision Notes

Summary

Algebraic expressions

Combining like terms and simplifying expressions

Multiplication and division operations

Factorising techniques

Common factor method

Difference of squares

Factorising trinomials

Notes on factorising trinomials of the form ax² + bx + c

Quadratic equations

Solving methods

Quadratic inequalities

Simultaneous equations

Solution methods

The nature of roots

Discriminant conditions

Remember!

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