Grade 10
Algebra, Functions, Finance, Trigonometry basics, Analytical Geometry, Statistics.
Quadratic Formula
Finds both roots of any quadratic equation ax² + bx + c = 0.
Use when factoring is not possible or when asked for exact roots.
x² − 5x + 6 = 0 → a=1, b=−5, c=6 → x = (5 ± 1)/2 → x = 3 or x = 2
Difference of Squares
Factorises any expression that is a perfect square minus a perfect square.
When you see two squared terms being subtracted.
x² − 9 = (x + 3)(x − 3)
Product Rule
When multiplying powers with the same base, add the exponents.
When multiplying exponential expressions with the same base.
2³ × 2⁴ = 2⁷ = 128
Quotient Rule
When dividing powers with the same base, subtract the exponents.
When dividing exponential expressions with the same base.
5⁶ ÷ 5² = 5⁴ = 625
Power of a Power
When raising a power to another power, multiply the exponents.
When you have an exponent raised to another exponent.
(2³)⁴ = 2¹² = 4096
Negative Exponent
A negative exponent means the reciprocal of the positive power.
When simplifying expressions with negative indices.
2⁻³ = 1/2³ = 1/8
Simple Interest
Calculates the total amount after simple interest is applied.
Short-term loans, hire purchase, and any scenario where interest does not compound.
R5000 at 8% for 3 years: A = 5000(1 + 0.08×3) = R6200
Simple Decay (Depreciation)
Calculates the value of an asset that depreciates at a simple rate.
Straight-line depreciation of vehicles, equipment, or assets.
R120 000 vehicle, 15% decay for 4 years: A = 120000(1 − 0.15×4) = R48 000
Sine Ratio
Ratio of the side opposite to the angle to the hypotenuse in a right triangle.
Finding an angle or side in any right-angled triangle.
sin 30° = 1/2 → opposite = 5 when hypotenuse = 10
Cosine Ratio
Ratio of the side adjacent to the angle to the hypotenuse.
Finding an angle or the adjacent side in a right triangle.
cos 60° = 0.5 → adjacent = 5 when hypotenuse = 10
Tangent Ratio
Ratio of the opposite side to the adjacent side.
When you know two sides of a right triangle (but not the hypotenuse).
tan 45° = 1 → opposite = adjacent = 7
Pythagorean Identity
The most fundamental trigonometric identity — true for all angles.
Simplifying trig expressions, proving identities, finding a trig ratio given another.
If sin θ = 3/5, then cos²θ = 1 − 9/25 = 16/25, so cos θ = 4/5
Distance Formula
Finds the straight-line distance between two points on the Cartesian plane.
Any problem involving the length between two coordinate points.
Between (1, 2) and (4, 6): d = √[(3)² + (4)²] = √25 = 5
Midpoint Formula
Finds the point exactly halfway between two given points.
Finding the centre of a line segment or the midpoint of a side in a shape.
Midpoint of (0, 0) and (6, 4) = (3, 2)
Gradient (Slope)
Measures the steepness and direction of a straight line.
Finding the slope of a line, checking if lines are parallel or perpendicular.
Between (1, 3) and (3, 7): m = (7 − 3)/(3 − 1) = 4/2 = 2
Perpendicular Gradients
Two lines are perpendicular if their gradients multiply to −1.
Proving perpendicularity or finding the gradient of a perpendicular line.
If m₁ = 3, then m₂ = −1/3 for a perpendicular line
Arithmetic Mean
The average of a data set — sum of all values divided by the count.
Describing the central tendency of a data set.
Data: {2, 4, 6, 8, 10} → x̄ = 30/5 = 6