Algebra is the skeleton key of Grade 12 Maths. Every single topic in Paper 1 depends on it.
Cannot factorise? You cannot do calculus. Cannot solve simultaneous equations? You cannot do functions. Cannot work with surds or exponents? You cannot do sequences.
Algebra carries about 25 marks directly in Paper 1 as Question 1. But indirectly, it touches another 100 marks across the rest of the paper. If your algebra is shaky, everything else wobbles.
This post goes beyond the basics. You already know how to solve x + 3 = 7. This is about the harder algebra that separates a level 3 from a level 5 and above.
In This Post You Will Learn
✓ How to solve quadratic equations using the formula, factorisation, and completing the square
✓ How to handle simultaneous equations where one is quadratic
✓ Surd equations and how to deal with extraneous solutions
✓ Inequality problems including quadratic inequalities
✓ Exponent equations that look impossible but follow a pattern
✓ How the NSC structures Question 1 and where the marks sit
Solving Quadratic Equations: Three Methods
You must be able to use all three methods. The exam does not tell you which one to use. You decide based on what the equation looks like.
Method 1: Factorisation
Use when the equation factorises neatly.
Example: x² - 5x + 6 = 0
(x - 2)(x - 3) = 0
x = 2 or x = 3
Method 2: Quadratic Formula
Use when factorisation is not obvious or when coefficients are ugly.
-b ± √(b² - 4ac)
x = ─────────────────────
2aExample: 2x² + 3x - 7 = 0
a = 2, b = 3, c = -7
x = [-3 ± √(9 + 56)] / 4
x = [-3 ± √65] / 4
x = [-3 + 8.06] / 4 or x = [-3 - 8.06] / 4
x = 1.27 or x = -2.77
Method 3: Completing the Square
Use when the question specifically asks for it, or when you need exact (surd) answers.
Example: x² + 6x + 2 = 0
Step 1: Move the constant.
x² + 6x = -2
Step 2: Take half the coefficient of x, square it, add to both sides.
Half of 6 = 3. Squared = 9.
x² + 6x + 9 = -2 + 9
Step 3: Write as a perfect square.
(x + 3)² = 7
Step 4: Square root both sides.
x + 3 = ±√7
x = -3 ± √7
| Method | When to Use |
|--------------------|------------------------------------------|
| Factorisation | Clean coefficients, obvious factors |
| Quadratic formula | Ugly numbers, no obvious factors |
| Completing square | Question asks for it, or exact answers |The discriminant tells you what to expect. b² - 4ac > 0 means two real solutions. b² - 4ac = 0 means one repeated root. b² - 4ac < 0 means no real solutions. The exam sometimes asks you to find values of k for which an equation has real/equal/no roots. Use the discriminant.
Simultaneous Equations: One Linear, One Quadratic
This appears in almost every Paper 1 Question 1.
The method is always substitution.
Example: Solve simultaneously.
y = 2x + 1 ... (1)
x² + y² = 25 ... (2)
Step 1: Substitute (1) into (2).
x² + (2x + 1)² = 25
Step 2: Expand.
x² + 4x² + 4x + 1 = 25
5x² + 4x - 24 = 0
Step 3: Factorise or use the formula.
(5x + 12)(x - 2) = 0
x = -12/5 or x = 2
Step 4: Substitute back into (1) to find y.
If x = 2: y = 2(2) + 1 = 5
If x = -12/5: y = 2(-12/5) + 1 = -24/5 + 5/5 = -19/5
Solutions: (2, 5) and (-12/5, -19/5)
Always substitute into the LINEAR equation to find y. Substituting into the quadratic creates extra solutions that you then have to check. The linear equation gives you one clean answer per x-value.
Surd Equations
A surd equation has a square root with x inside it.
Example: √(x + 5) = x - 1
Step 1: Square both sides.
x + 5 = (x - 1)²
x + 5 = x² - 2x + 1
Step 2: Rearrange to standard form.
0 = x² - 3x - 4
0 = (x - 4)(x + 1)
x = 4 or x = -1
Step 3: CHECK both solutions in the original equation.
x = 4: √(4 + 5) = √9 = 3 and 4 - 1 = 3. ✓ Valid.
x = -1: √(-1 + 5) = √4 = 2 and -1 - 1 = -2. ✗ Invalid. (2 ≠ -2)
Only x = 4 is valid.
You MUST check surd solutions. Squaring both sides can introduce extraneous solutions that do not actually satisfy the original equation. If you do not check, you might include a wrong answer and lose marks.
For full live lessons on algebra and all Paper 1 topics, see our Grade 12 Maths tuition page.
Quadratic Inequalities
Quadratic inequalities are solved differently from equations. You cannot just factorise and stop. You need to determine the intervals.
Example: x² - 5x + 6 > 0
Step 1: Factorise.
(x - 2)(x - 3) > 0
Step 2: Find the critical values.
x = 2 and x = 3
Step 3: Draw a number line and test intervals.
----+--------+----
2 3
Test x = 0: (0-2)(0-3) = (-2)(-3) = +6 > 0 ✓
Test x = 2.5: (0.5)(-0.5) = -0.25 < 0 ✗
Test x = 4: (2)(1) = +2 > 0 ✓Step 4: Write the solution.
x < 2 or x > 3
Parabola shortcut: If a > 0 (happy face parabola) and you want > 0, the solution is the two outer regions. If you want < 0, the solution is the region between the roots. For a sad face parabola (a < 0), it is the opposite.
If you want to see how inequalities connect to graph reading, check out Grade 12 Functions and Graphs - Everything You Need to Know.
Exponent Equations
These look scary but always follow the same principle: get the bases the same.
Example: 3^(2x+1) = 81
Step 1: Write 81 as a power of 3.
81 = 3⁴
Step 2: Now the bases are equal, so set the exponents equal.
2x + 1 = 4
2x = 3
x = 3/2
Harder Example: Different Bases
Example: 2^x = 5
You cannot make the bases the same. Use logarithms.
x = log₂5 = log5/log2 = 0.699/0.301 = 2.32
Exponential Equations With Substitution
Example: 4^x - 3(2^x) + 2 = 0
Step 1: Notice that 4^x = (2²)^x = (2^x)².
Step 2: Let k = 2^x.
k² - 3k + 2 = 0
(k - 1)(k - 2) = 0
k = 1 or k = 2
Step 3: Substitute back.
2^x = 1, so x = 0
2^x = 2, so x = 1
| Exponent Equation Type | Method |
|-----------------------------|-----------------------------------|
| Same base possible | Equal bases, equal exponents |
| Cannot get same base | Use logarithms |
| Quadratic in exponential form| Substitute k = base^x, solve quadratic |Nature of Roots (Discriminant)
The discriminant determines the nature of the roots of ax² + bx + c = 0.
Δ = b² - 4ac
| Value of Δ | Nature of Roots |
|--------------|------------------------------------------|
| Δ > 0 | Two distinct real roots |
| Δ = 0 | Two equal real roots (one repeated root) |
| Δ < 0 | No real roots (non-real/imaginary) |
| Δ ≥ 0 | Real roots |
| Δ > 0 and | |
| Δ is a | Rational roots |
| perfect square| |Example: For which values of k does x² + 4x + k = 0 have real roots?
Δ ≥ 0
b² - 4ac ≥ 0
16 - 4(1)(k) ≥ 0
16 - 4k ≥ 0
4k ≤ 16
k ≤ 4
If you want to avoid the common errors that cost marks in algebra, read 10 Most Common Mistakes in Grade 12 Maths Paper 1.
Common Mistakes Students Make
- Not checking surd equation solutions
Squaring introduces extraneous roots. You must substitute back into the original equation and verify. Skip this and you either include a wrong answer or lose the "verification" mark.
- Stopping at the critical values for inequalities
x² - 5x + 6 > 0 does NOT give you x > 2 and x > 3. Those are the critical values, not the solution. You must test intervals on a number line to determine the actual solution set.
- Substituting back into the quadratic for simultaneous equations
Always substitute into the linear equation. Substituting into the quadratic gives you extra solutions and wastes time. If you cannot find a linear equation, check your setup.
- Forgetting that 2^x is always positive
When you solve 2^x = -3, there is no solution. An exponential with a positive base never equals a negative number. Students who do not know this try to use logarithms on a negative number and get confused.
- Writing the discriminant condition wrong
For real roots, Δ ≥ 0. For non-real roots, Δ < 0. For equal roots, Δ = 0. Students sometimes flip the inequality. Know these conditions cold.
How This Topic Appears in the NSC Exam
Algebra appears in Paper 1 as Question 1, the very first question in the paper.
It carries approximately 25 marks and is split into several sub-questions covering different equation types.
| Typical Question 1 Breakdown | Marks |
|-------------------------------------|-------|
| Quadratic equation (factorise) | 3 |
| Quadratic equation (formula/surds) | 3-4 |
| Simultaneous equations | 5-6 |
| Quadratic inequality | 4 |
| Surd equation | 4-5 |
| Nature of roots (discriminant) | 4-5 |
| TOTAL | ~25 |In the 2023 NSC exam, Question 1 included a factorisable quadratic, a simultaneous equation with one linear and one quadratic, a quadratic inequality, and a discriminant question involving an unknown parameter k.
The question order goes from easier to harder. Parts (a) and (b) are typically straightforward factorisation or formula questions. Parts (c) through (f) get progressively harder with simultaneous equations, surds, and discriminant problems.
Strategy: Do Question 1 quickly and accurately. It sets the tone for the paper. If you can pick up 20+ out of 25 in the first 20 minutes, you go into the rest of Paper 1 with confidence and marks in the bank.
For the full Paper 1 structure, read
https://agameacademy.co.za/blogs/study-guide/nsc-maths-exam-format-explained-paper-1-vs-paper-2
Want live lessons covering algebra and every Paper 1 topic?
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