🧮 Math Formulas
61 formulas in 13 sections. For each one you get the formula, what every symbol means, where it is used, an explanation in plain words and a worked example.
Sections: Identities · Linear Equations · Quadratic Equations · Arithmetic Progression · Geometric Progression · Exponents · Logarithms · Series (Sum of n terms) · Polynomials · Inequalities · Ratio & Proportion · Harmonic Progression · Surds
(a + b)² = a² + 2ab + b²
| a, b | any numbers or terms |
|---|
Where it is used: To expand or quickly square a sum, and to simplify algebraic expressions and mental-maths squaring.
In simple words: Squaring a two-part sum gives each part squared plus twice their product.
Example: (x+3)² = x² + 6x + 9
(a − b)² = a² − 2ab + b²
| a, b | any numbers or terms |
|---|
Where it is used: To expand the square of a difference in algebra and to square numbers like 99 easily.
In simple words: Squaring a two-part difference gives each part squared minus twice their product.
Example: (x−2)² = x² − 4x + 4
a² − b² = (a + b)(a − b)
| a, b | any numbers or terms |
|---|
Where it is used: To factorise a difference of squares — very common in simplification and solving equations.
In simple words: One square minus another equals the sum times the difference of the two numbers.
Example: x² − 9 = (x+3)(x−3)
(x + a)(x + b) = x² + (a + b)x + ab
| x | the variable |
|---|---|
| a, b | constants |
Where it is used: To expand the product of two simple binomials and to help factorise quadratics.
In simple words: Multiply out the brackets: the middle number is the sum of a and b, the last is their product.
Example: (x+2)(x+3) = x² + 5x + 6
(a + b)³ = a³ + 3a²b + 3ab² + b³
| a, b | any numbers or terms |
|---|
Where it is used: To expand the cube of a sum in algebra and to cube numbers quickly.
In simple words: The cube of a two-part sum expands into four terms following the 1-3-3-1 pattern.
Example: (x+1)³ = x³ + 3x² + 3x + 1
(a − b)³ = a³ − 3a²b + 3ab² − b³
| a, b | any numbers or terms |
|---|
Where it is used: To expand the cube of a difference in algebra.
In simple words: Like the cube of a sum, but the signs alternate + − + −.
Example: (x−1)³ = x³ − 3x² + 3x − 1
a³ + b³ = (a + b)(a² − ab + b²)
| a, b | any numbers or terms |
|---|
Where it is used: To factorise a sum of cubes in simplification and equation-solving.
In simple words: Splits a sum of two cubes into a simple bracket times a longer one.
Example: x³ + 8 = (x+2)(x²−2x+4)
a³ − b³ = (a − b)(a² + ab + b²)
| a, b | any numbers or terms |
|---|
Where it is used: To factorise a difference of cubes in algebra.
In simple words: Splits a difference of two cubes into a simple bracket times a longer one.
Example: x³ − 27 = (x−3)(x²+3x+9)
(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
| a, b, c | any three terms |
|---|
Where it is used: To expand the square of a three-term sum in algebra.
In simple words: Square each of the three parts, then add twice each pair's product.
Example: (x+y+1)² = x²+y²+1+2xy+2y+2x
a² + b² = (a + b)² − 2ab
| a, b | any numbers or terms |
|---|
Where it is used: To find a²+b² quickly when the sum and product of a and b are known.
In simple words: Square the sum and subtract twice the product to get the sum of the squares.
Example: a+b=5, ab=6 → a²+b² = 25 − 12 = 13
(a+b)² − (a−b)² = 4ab
| a, b | any terms |
|---|
Where it is used: A handy identity to find 4ab or ab from the two squares.
In simple words: The square of the sum minus the square of the difference is always four times their product.
Example: (5+3)²−(5−3)² = 64−4 = 60 = 4·15
(a+b)² + (a−b)² = 2(a² + b²)
| a, b | any terms |
|---|
Where it is used: To find a²+b² from the sum and difference squares.
In simple words: Add the two squares and you get twice the sum of the individual squares.
Example: (5+3)²+(5−3)² = 64+4 = 68 = 2·34
a³+b³+c³ − 3abc = (a+b+c)(a²+b²+c² − ab − bc − ca)
| a, b, c | any three terms |
|---|
Where it is used: To factorise expressions with three cubes — common in algebra proofs.
In simple words: A factorising rule for the sum of three cubes minus three times their product.
Example: If a+b+c=0, then a³+b³+c³ = 3abc
(a + b)(a − b) = a² − b²
| a, b | any numbers |
|---|
Where it is used: A fast way to multiply numbers like 52×48 = (50+2)(50−2).
In simple words: Sum times difference of two numbers equals the difference of their squares.
Example: 52×48 = 50²−2² = 2500−4 = 2496
ax + b = c ⇒ x = (c − b) ÷ a
| a | coefficient of x |
|---|---|
| b, c | constants |
| x | the unknown |
Where it is used: To find an unknown value in a one-variable equation — the basis of all algebra word problems.
In simple words: Move the added number to the other side, then divide by the number in front of x to find x.
Example: 2x + 3 = 11 → x = (11−3)/2 = 4
ax + by + c = 0
| a, b | coefficients (not both 0) |
|---|---|
| c | constant |
| x, y | the two unknowns |
Where it is used: The standard form of a straight-line equation; solved in pairs by substitution or elimination.
In simple words: An equation with two unknowns; its solutions are all the points that lie on a straight line.
Example: 2x + 3y − 6 = 0 passes through (3,0) and (0,2)
a₁/a₂ ≠ b₁/b₂ (one sol.); = but ≠ c₁/c₂ (none); all equal (infinite)
| a, b, c | coefficients of the two equations |
|---|
Where it is used: To decide whether a pair of linear equations has one, none, or infinitely many solutions.
In simple words: Compare the three ratios: they tell you if the lines cross, are parallel, or are the same line.
Example: x+y=2 and 2x+2y=4 → all ratios equal → infinite solutions
x = (−b ± √(b² − 4ac)) ÷ 2a
| a, b, c | coefficients of ax²+bx+c = 0 |
|---|---|
| x | the roots (solutions) |
Where it is used: To solve any quadratic equation ax²+bx+c = 0 — projectile motion, areas, optimisation.
In simple words: A ready-made recipe that gives the two answers of a squared equation from its three numbers.
Example: x²−5x+6=0 → x = (5±1)/2 = 3 or 2
D = b² − 4ac
| a, b, c | coefficients |
|---|---|
| D | the discriminant |
Where it is used: To find how many real solutions a quadratic has without fully solving it (D>0 two, D=0 one, D<0 none).
In simple words: This one number tells you whether the equation has two answers, one, or none, before you solve it.
Example: x²−5x+6: D = 25−24 = 1 (>0, two real roots)
α + β = −b/a, α × β = c/a
| α, β | the two roots |
|---|---|
| a, b, c | coefficients |
Where it is used: To find or check the roots of a quadratic, and to build a quadratic from its roots.
In simple words: Without solving, the two answers add up to −b/a and multiply to c/a.
Example: x²−5x+6: sum = 5, product = 6 (roots 2 and 3)
ax² + bx + c = 0, a ≠ 0
| a | coefficient of x² |
|---|---|
| b | coefficient of x |
| c | constant term |
Where it is used: The general shape of every quadratic equation, the starting point for solving them.
In simple words: Any equation with an x² term (and no higher power) can be written this way before solving.
Example: x² − 5x + 6 = 0 has a=1, b=−5, c=6
D>0: two real, D=0: one real, D<0: no real roots
| D | b² − 4ac (the discriminant) |
|---|
Where it is used: To decide how many real solutions a quadratic has before solving it.
In simple words: The sign of that one number D tells you if there are two answers, one, or none.
Example: x²+1=0: D = −4 < 0 → no real roots
x² + bx = (x + b/2)² − (b/2)²
| b | coefficient of x |
|---|
Where it is used: To rewrite a quadratic as a perfect square — used to derive the quadratic formula and find vertices.
In simple words: Turn a quadratic into a squared bracket plus a leftover number.
Example: x²+6x = (x+3)² − 9
α − β = √D ÷ a, D = b² − 4ac
| α, β | the roots |
|---|---|
| D | discriminant |
| a | coefficient of x² |
Where it is used: To find how far apart the two roots of a quadratic are.
In simple words: The gap between the two answers comes from the square root of the discriminant.
Example: x²−5x+6: D=1, a=1 → α−β = 1
x = −b/2a, then y = value at that x
| a, b | coefficients |
|---|---|
| (x, y) | the turning point |
Where it is used: To find the highest or lowest point of a quadratic — maxima/minima, projectile peaks.
In simple words: The turning point of the U-shaped curve sits at x = −b/(2a).
Example: y=x²−4x+3: vertex at x = 2, y = −1
x = −b ÷ 2a
| a, b | coefficients of the quadratic |
|---|
Where it is used: The vertical line that splits a parabola into two mirror halves.
In simple words: The U-shape is symmetric about the vertical line x = −b/(2a).
Example: y=x²−6x+5: axis is x = 3
aₙ = a + (n − 1)d
| a | first term |
|---|---|
| d | common difference |
| n | term number |
Where it is used: To find any term of a sequence that increases by a fixed step — savings, seating rows, patterns.
In simple words: Start with the first number and add the fixed step (n−1) times to reach the nth number.
Example: a=2, d=3: a₅ = 2 + 4×3 = 14
Sₙ = n/2 × [2a + (n − 1)d]
| a | first term |
|---|---|
| d | common difference |
| n | number of terms |
Where it is used: To add up many equally-spaced numbers quickly — totals of savings, stacked logs, number series.
In simple words: Instead of adding one by one, this finds the total of an evenly-stepping list in one go.
Example: a=2, d=3, n=5: S = 5/2×(4+12) = 40
Sₙ = n/2 × (a + l)
| a | first term |
|---|---|
| l | last term |
| n | number of terms |
Where it is used: A quick way to add an AP when the first and last terms are known.
In simple words: Average the first and last numbers, then multiply by how many numbers there are.
Example: a=2, l=14, n=5: S = 5/2×(2+14) = 40
AM = (a + b) ÷ 2
| a, b | the two numbers |
|---|
Where it is used: To find the middle term that sits in AP between two numbers.
In simple words: The arithmetic mean is just the ordinary average of the two numbers.
Example: AM of 4 and 10 = 7
a − d, a, a + d
| a | middle term |
|---|---|
| d | common difference |
Where it is used: A convenient way to represent three unknown terms of an AP when solving problems.
In simple words: Write three evenly-spaced numbers around a middle value to make the algebra simple.
Example: For sum 15: (5−d), 5, (5+d) sum to 15
aₙ = a × r^(n−1)
| a | first term |
|---|---|
| r | common ratio |
| n | term number |
Where it is used: To find any term of a sequence that multiplies by a fixed factor — compound growth, populations.
In simple words: Start with the first number and keep multiplying by the fixed ratio (n−1) times.
Example: a=3, r=2: a₄ = 3 × 2³ = 24
Sₙ = a(rⁿ − 1) ÷ (r − 1), r ≠ 1
| a | first term |
|---|---|
| r | common ratio |
| n | number of terms |
Where it is used: To add up a multiplying sequence — total growth, repeated doubling, loan/interest sums.
In simple words: Adds a list where each number is a fixed multiple of the one before, all at once.
Example: a=3, r=2, n=4: S = 3(16−1)/1 = 45
GM = √(a × b)
| a, b | the two positive numbers |
|---|
Where it is used: To find the middle term that sits in GP between two numbers — growth rates, averages.
In simple words: The geometric mean is the square root of the product of the two numbers.
Example: GM of 4 and 9 = √36 = 6
S∞ = a ÷ (1 − r), |r| < 1
| a | first term |
|---|---|
| r | common ratio (between −1 and 1) |
Where it is used: To add a never-ending GP whose terms keep shrinking — used in recurring decimals and limits.
In simple words: When each term is a small fraction of the last, even an endless list adds to a finite total.
Example: 1 + ½ + ¼ + … = 1/(1−½) = 2
a/r, a, a × r
| a | middle term |
|---|---|
| r | common ratio |
Where it is used: A convenient way to represent three unknown terms of a GP when solving problems.
In simple words: Write three numbers each a fixed multiple of the middle one to simplify the algebra.
Example: For product 27: (3/r), 3, 3r multiply to 27
If a₁x+b₁y=c₁ and a₂x+b₂y=c₂, unique solution when a₁/a₂ ≠ b₁/b₂
| a, b, c | the coefficients of the two equations |
|---|
Where it is used: To check whether a pair of linear equations has one solution, none, or infinitely many.
In simple words: Compare the ratios of the numbers: if the x and y ratios differ, the two lines cross at exactly one point.
Example: 2x+3y=6 and 4x+9y=15 → 2/4 ≠ 3/9, so one solution
(ab)ⁿ = aⁿbⁿ, (a/b)ⁿ = aⁿ/bⁿ
| a, b | the bases |
|---|---|
| n | the power |
Where it is used: To distribute a power across a product or a fraction in algebra.
In simple words: A power on a bracket spreads to each factor inside.
Example: (2x)³ = 8x³
log(m × n) = log m + log n
| m, n | positive numbers |
|---|
Where it is used: To turn a multiplication into an addition of logs — used in calculation and calculus.
In simple words: The log of a product is the sum of the logs.
Example: log(2×5) = log2 + log5 = log10 = 1
log(m ÷ n) = log m − log n
| m, n | positive numbers |
|---|
Where it is used: To turn a division into a subtraction of logs.
In simple words: The log of a fraction is the difference of the logs.
Example: log(100/10) = log100 − log10 = 2 − 1 = 1
log(mⁿ) = n × log m
| m | positive number |
|---|---|
| n | the power |
Where it is used: To bring a power down in front of a log — used to solve exponential equations.
In simple words: The log of a power moves the little top number in front as a multiplier.
Example: log(10³) = 3 × log10 = 3
log_b a = log a ÷ log b
| a | the number |
|---|---|
| b | the base |
Where it is used: To evaluate a log in any base using common (base-10) or natural logs.
In simple words: Any log can be worked out by dividing two ordinary logs.
Example: log₂8 = log8 / log2 = 3
log_a a = 1, log_a 1 = 0
| a | the base (a>0, a≠1) |
|---|
Where it is used: Two standard log values used constantly in simplification.
In simple words: The log of the base itself is 1, and the log of 1 is always 0.
Example: log₅5 = 1, log₅1 = 0
log(ⁿ√m) = (1/n) × log m
| m | positive number |
|---|---|
| n | the root |
Where it is used: To take the log of a root by turning the root into a fractional power.
In simple words: The log of an nth root is the log divided by n.
Example: log(√100) = ½ × log100 = 1
if log_a x = y, then x = aʸ
| a | the base |
|---|---|
| x | the number |
| y | the log |
Where it is used: The basic definition linking logarithms and exponents — used to solve exponential equations.
In simple words: A logarithm just asks 'what power gives this number?'.
Example: log₂8 = 3 because 2³ = 8
1 + 2 + 3 + … + n = n(n+1) ÷ 2
| n | how many terms |
|---|
Where it is used: To add all whole numbers up to n instantly — counting, patterns, AP.
In simple words: Instead of adding 1 to 100 one by one, this gives the total at once.
Example: 1+…+100 = 100·101/2 = 5050
1 + 3 + 5 + … = n²
| n | how many odd numbers |
|---|
Where it is used: A neat result: the sum of the first n odd numbers is a perfect square.
In simple words: Add the first n odd numbers and you always get n squared.
Example: 1+3+5+7 = 16 = 4²
1² + 2² + … + n² = n(n+1)(2n+1) ÷ 6
| n | how many terms |
|---|
Where it is used: To add the squares of the first n numbers — used in statistics and calculus.
In simple words: A shortcut for the total of 1², 2², 3², … up to n².
Example: 1²+2²+3² = 3·4·7/6 = 14
1³ + 2³ + … + n³ = [n(n+1) ÷ 2]²
| n | how many terms |
|---|
Where it is used: To add the cubes of the first n numbers.
In simple words: The sum of the first n cubes equals the square of the sum of the first n numbers.
Example: 1³+2³+3³ = 36 = (6)²
2 + 4 + 6 + … = n(n + 1)
| n | how many even numbers |
|---|
Where it is used: To add the first n even numbers instantly.
In simple words: Add the first n even numbers and you get n times (n+1).
Example: 2+4+6+8 = 4×5 = 20
Remainder of p(x) ÷ (x − a) = p(a)
| p(x) | the polynomial |
|---|---|
| a | the value making the divisor 0 |
Where it is used: To find the remainder of a polynomial division without doing the division.
In simple words: Just put x = a into the polynomial; the answer is the remainder.
Example: p(x)=x²+2x+1, ÷(x−1): remainder = p(1) = 4
(x − a) is a factor of p(x) ⇔ p(a) = 0
| p(x) | the polynomial |
|---|---|
| a | a possible root |
Where it is used: To test whether (x−a) divides a polynomial exactly, and to factorise polynomials.
In simple words: If putting x = a makes the polynomial zero, then (x−a) is a factor of it.
Example: p(x)=x²−4, p(2)=0 → (x−2) is a factor
Sum = −b/a, Product = c/a for ax²+bx+c
| a, b, c | coefficients |
|---|
Where it is used: To relate the roots of a quadratic to its coefficients, or build a quadratic from roots.
In simple words: The two answers add to −b/a and multiply to c/a.
Example: x²−7x+12: sum=7, product=12 (roots 3,4)
x² − (sum)x + (product) = 0
| sum | α+β |
|---|---|
| product | α·β |
Where it is used: To write a quadratic equation when its two roots (or their sum and product) are known.
In simple words: Put the sum and product of the two answers into this pattern to get the equation.
Example: Roots 2,3 → x² − 5x + 6 = 0
p(x) = g(x) × q(x) + r(x)
| p(x) | dividend |
|---|---|
| g(x) | divisor |
| q(x) | quotient |
| r(x) | remainder |
Where it is used: The rule behind polynomial long division — used to factorise and simplify.
In simple words: Just like numbers: dividend equals divisor times quotient plus remainder.
Example: x²−1 = (x−1)(x+1) + 0
α+β+γ = −b/a, αβ+βγ+γα = c/a, αβγ = −d/a
| a,b,c,d | coefficients of ax³+bx²+cx+d |
|---|---|
| α,β,γ | the three roots |
Where it is used: To relate the three roots of a cubic to its coefficients.
In simple words: For a cubic, the three answers add, pair-multiply and all-multiply to fixed values from the coefficients.
Example: x³−6x²+11x−6: sum=6, product=6
If a > b and c < 0, then ac < bc
| a, b | the numbers |
|---|---|
| c | a negative multiplier |
Where it is used: The key rule when solving inequalities — the inequality sign reverses.
In simple words: Multiplying or dividing an inequality by a negative number flips the < or > sign.
Example: −2x > 6 → x < −3 (sign flipped)
If a : b = c : d, then d = (b × c) ÷ a
| a, b, c | the three known quantities |
|---|---|
| d | the fourth |
Where it is used: To find the missing fourth term in a proportion — scaling recipes, maps, prices.
In simple words: Cross-multiply to find the missing fourth number that keeps the ratio.
Example: 2:3 = 4:d → d = 3×4/2 = 6
If a : x = x : b, then x = √(a × b)
| a, b | the two numbers |
|---|---|
| x | the mean proportional |
Where it is used: To find the number that is in continued proportion between two numbers.
In simple words: The middle term of a continued proportion is the square root of the product.
Example: 4 : x = x : 9 → x = √36 = 6
HM = 2ab ÷ (a + b)
| a, b | the two numbers |
|---|
Where it is used: To average rates like speeds over equal distances, and in HP problems.
In simple words: The harmonic mean is used when averaging rates — twice the product over the sum.
Example: HM of 4 and 6 = 2·24/10 = 4.8
1/√a = √a ÷ a
| a | positive number under the root |
|---|
Where it is used: To remove a surd from the denominator so the fraction is easier to work with.
In simple words: Multiply top and bottom by the root to clear the root from the bottom.
Example: 1/√2 = √2/2 ≈ 0.707