Grade 12 formula & reference bank

x = (-b ± √(b² - 4ac)) / 2a

a, b, c

coefficients of ax² + bx + c = 0

x

the roots (solutions)

When to use it: Any quadratic equation that does not factorise easily.

Write the equation in standard form first, then substitute a, b and c. Work out the discriminant before dividing.

Example: 2x² - 5x + 1 = 0 gives x = (5 ± √17)/4.

Δ = b² - 4ac

Δ

discriminant

When to use it: When a question asks about the nature of the roots without solving.

Δ > 0 two real roots; Δ = 0 one real root; Δ < 0 non-real roots. A perfect square Δ means rational roots.

Example: For x² - 6x + 9 = 0, Δ = 0, so the roots are real and equal.

Tₙ = a + (n - 1)d

a

first term

d

common difference

n

term number

When to use it: A sequence with a constant difference between terms.

Check T₂ - T₁ = T₃ - T₂ before using it.

Sₙ = n/2 (2a + (n - 1)d) = n/2 (a + l)

Sₙ

sum of n terms

l

last term

When to use it: Adding a fixed number of terms of an arithmetic sequence.

Use the (a + l) version when you already know the last term.

Tₙ = a·r^(n-1)

r

constant ratio

When to use it: A sequence where each term is multiplied by the same ratio.

Find r by dividing any term by the one before it.

Sₙ = a(rⁿ - 1)/(r - 1), r ≠ 1

Sₙ

sum of n terms

When to use it: Adding n terms of a geometric sequence.

Use a(1 - rⁿ)/(1 - r) when r < 1 to keep the numbers positive.

S∞ = a/(1 - r), -1 < r < 1

S∞

sum to infinity

When to use it: A converging geometric series only.

State the condition -1 < r < 1 in your answer; without it the series diverges.

∑(n=1 to k) Tₙ

∑

sum of

k

last value of n

When to use it: Writing a series compactly, or reading one from a paper.

Substitute the lower and upper values of n to see the first and last terms, then choose the arithmetic or geometric formula.

A = P(1 + in)

A

accumulated amount

P

principal

i

interest rate per period as a decimal

n

number of periods

When to use it: Interest calculated on the original amount only.

Convert a percentage to a decimal first: 9,5% is 0,095.

A = P(1 + i)ⁿ

A

accumulated amount

i

rate per compounding period

n

number of compounding periods

When to use it: Growth where interest earns interest.

Match i and n to the compounding period: monthly at 12% p.a. means i = 0,12/12 and n = months.

A = P(1 - i)ⁿ (reducing balance)

A

book value

i

depreciation rate

When to use it: Reducing-balance depreciation of an asset.

Straight-line depreciation uses A = P(1 - in) instead.

1 + i(eff) = (1 + i(nom)/m)^m

m

compounding periods per year

When to use it: Comparing rates advertised with different compounding periods.

The effective rate is what you actually pay or earn in a year.

F = x[(1 + i)ⁿ - 1]/i

F

future value

x

regular payment

When to use it: Regular savings, e.g. a retirement annuity.

The first payment must be at the end of the first period; adjust n if it is not.

P = x[1 - (1 + i)^-n]/i

P

loan amount

x

regular repayment

When to use it: Loans and bonds where repayments are equal.

Use it to find the monthly repayment on a home loan or the outstanding balance.

f'(x) = lim(h→0) [f(x + h) - f(x)]/h

h

a small change in x

f'(x)

the derivative

When to use it: When the question says “from first principles”.

Write the limit each line until the very last step; simplify the numerator before dividing by h.

d/dx[axⁿ] = anx^(n-1)

n

any real exponent

When to use it: Differentiating polynomials and terms rewritten as powers.

Rewrite roots and fractions as powers first: √x = x^(1/2).

y - y₁ = f'(x₁)(x - x₁)

f'(x₁)

gradient at the point of contact

When to use it: Finding the tangent to a curve at a point.

The gradient of the tangent is the derivative evaluated at that x-value.

f'(x) = 0

f'(x)

gradient function

When to use it: Local maximum, local minimum or point of inflection.

Solve f'(x) = 0 for x, then test the sign of f' on either side or use f''(x).

f''(x) = 0

f''(x)

second derivative

When to use it: Sketching cubic graphs and describing concavity.

Concavity changes there: f'' < 0 is concave down, f'' > 0 is concave up.

rate = dy/dx (or ds/dt for motion)

s

displacement

t

time

When to use it: Word problems about speed, growth or optimisation.

Velocity is the derivative of displacement; acceleration is the derivative of velocity.

∫axⁿ dx = ax^(n+1)/(n+1) + C, n ≠ -1

C

constant of integration

When to use it: Reversing differentiation and finding areas under simple curves.

Never forget + C on an indefinite integral.

P(A or B) = P(A) + P(B) - P(A and B)

P(A)

probability of A

When to use it: Any two events, overlapping or not.

For mutually exclusive events P(A and B) = 0.

P(A and B) = P(A) × P(B)

P(A and B)

both events happen

When to use it: Testing or using independence.

If the equation does not hold, the events are dependent.

P(not A) = 1 - P(A)

not A

A does not happen

When to use it: “At least one” questions.

P(at least one) = 1 - P(none).

n = n₁ × n₂ × ... × nₖ

nᵢ

choices at each stage

When to use it: Arrangements, codes and number plates.

Use n! for arrangements of all items and treat items that must stay together as one block.

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

d

length of the line segment

When to use it: Lengths of sides, radii and proving shapes.

Squaring removes the sign, so the order of the points does not matter.

m = (y₂ - y₁)/(x₂ - x₁)

m

gradient

When to use it: Parallel (m₁ = m₂) and perpendicular (m₁ × m₂ = -1) lines.

A vertical line has an undefined gradient.

M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

M

midpoint of the segment

When to use it: Diagonals of parallelograms and centres of circles.

Average the x-values and the y-values separately.

(x - a)² + (y - b)² = r²

(a, b)

centre

r

radius

When to use it: Circle geometry in the Cartesian plane.

Complete the square to change x² + y² + Dx + Ey + F = 0 into this form.

m = tanθ

θ

angle the line makes with the positive x-axis

When to use it: Angles between lines.

If tanθ is negative, add 180° so that 0° ≤ θ < 180°.

sin(A ± B) = sinAcosB ± cosAsinB; cos(A ± B) = cosAcosB ∓ sinAsinB

A, B

any two angles

When to use it: Simplifying or proving expressions with sums of angles.

Watch the sign swap in the cosine identity.

sin2A = 2sinAcosA; cos2A = cos²A - sin²A = 1 - 2sin²A = 2cos²A - 1

2A

double the angle

When to use it: Reducing 2A to A, or building 2A from A.

Choose the version of cos2A that matches the other terms in the expression.

tanθ = sinθ/cosθ; sin²θ + cos²θ = 1

θ

any angle

When to use it: Almost every trig proof starts here.

Turn everything into sines and cosines when you get stuck.

a/sinA = b/sinB = c/sinC

a, b, c

sides opposite angles A, B, C

When to use it: Two angles and a side, or two sides and a non-included angle.

Watch for the ambiguous case when finding an angle.

a² = b² + c² - 2bc·cosA

A

angle opposite side a

When to use it: Two sides and the included angle, or all three sides.

Rearrange to cosA = (b² + c² - a²)/2bc to find an angle.

Area = ½·ab·sinC

C

included angle

When to use it: Area of a triangle without a perpendicular height.

The angle must be between the two sides you use.

sinθ = k → θ = sin⁻¹k + 360°n or 180° - sin⁻¹k + 360°n; cosθ = k → θ = ±cos⁻¹k + 360°n; tanθ = k → θ = tan⁻¹k + 180°n

n

any integer

When to use it: Trig equations without a restricted domain.

Give the general solution first, then substitute integers for n to fit a given interval.

O

centre of the circle

When to use it: Proofs and riders in Paper 2.

Angle at centre = 2 × angle at circumference; angles in the same segment are equal; opposite angles of a cyclic quadrilateral add to 180°; tan-chord angle equals the angle in the alternate segment; a tangent is perpendicular to the radius at the point of contact.

Example: Always quote the reason for each statement — marks are awarded for reasons.

DE ∥ BC ⇒ AD/DB = AE/EC

∥

is parallel to

When to use it: Proportion theorem and similar triangles.

Similar triangles (equiangular) have sides in proportion; the ratio of their areas is the square of the ratio of the sides.

Cylinder: V = πr²h, SA = 2πr² + 2πrh; Cone: V = ⅓πr²h; Sphere: V = 4/3·πr³, SA = 4πr²

r

radius

h

perpendicular height

When to use it: Solids in Paper 2 measurement questions.

Check whether the question wants total surface area or only the curved surface.

σ = √[∑(x - x̄)²/n]

σ

standard deviation

x̄

mean

n

number of data values

When to use it: Describing spread around the mean.

Use your calculator's statistics mode in the exam; show the mean and n.

ŷ = A + Bx; r ∈ [-1, 1]

ŷ

predicted value

r

correlation coefficient

When to use it: Bivariate data and scatter plots.

|r| close to 1 is a strong linear relationship; r near 0 is weak. Never extrapolate far beyond the data.

Q₁, Q₂, Q₃

quartiles

When to use it: Cumulative frequency graphs and box-and-whisker plots.

Plot cumulative frequency against the upper boundary of each interval; read the median at half the total frequency.

Example: Interquartile range = Q₃ - Q₁.