Mathematics - StudyReady

Foundations

Algebra formula sheet

Concept	Formula
Quadratic formula	`x = (-b ± √(b² - 4ac)) / 2a`, for `ax² + bx + c = 0`
Slope of a line	`m = (y² - y¹) / (x² - x¹)`
Slope-intercept form	`y = mx + b`
Point-slope form	`y - y¹ = m(x - x¹)`
Difference of squares	`a² - b² = (a + b)(a - b)`
Perfect square trinomial	`(a ± b)² = a² ± 2ab + b²`
Exponent rules	`a^m·aⁿ = a^m+n`, `(a^m)ⁿ = a^mn`, `a⁰ = 1`
Logarithm identity	`log_b(x) = y` means `b^y = x`

Worked example: solve 2x² + 3x - 2 = 0. Here a=2, b=3, c=-2. Plug into the quadratic formula: x = (-3 ± √(9 + 16)) / 4 = (-3 ± 5) / 4, giving x = 0.5 or x = -2.

Shapes & Space

Geometry formula sheet

Shape	Area	Perimeter / Circumference
Rectangle	`A = lw`	`P = 2(l + w)`
Triangle	`A = ½bh`	`P = a + b + c`
Circle	`A = πr²`	`C = 2πr`
Trapezoid	`A = ½(b¹ + b²)h`	-

Solid	Volume	Surface Area
Rectangular prism	`V = lwh`	`SA = 2(lw + lh + wh)`
Cylinder	`V = πr²h`	`SA = 2πrh + 2πr²`
Sphere	`V = &frac43;πr³`	`SA = 4πr²`
Cone	`V = ⅓πr²h`	-

Pythagorean theorem: for a right triangle with legs a, b and hypotenuse c, a² + b² = c².

Angles

Trigonometry reference

Ratio	Definition (right triangle)
Sine	`sin(θ) = opposite / hypotenuse`
Cosine	`cos(θ) = adjacent / hypotenuse`
Tangent	`tan(θ) = opposite / adjacent`
Pythagorean identity	`sin²(θ) + cos²(θ) = 1`
Law of sines	`a/sin(A) = b/sin(B) = c/sin(C)`
Law of cosines	`c² = a² + b² - 2ab·cos(C)`

Memory trick: SOH-CAH-TOA - Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

Change & Accumulation

Calculus quick reference

Rule	Formula
Power rule (derivative)	`d/dx[xⁿ] = nx^n-1`
Product rule	`d/dx[fg] = f'g + fg'`
Quotient rule	`d/dx[f/g] = (f'g - fg') / g²`
Chain rule	`d/dx[f(g(x))] = f'(g(x))·g'(x)`
Power rule (integral)	`∫xⁿdx = xⁿ⁺¹/(n+1) + C`
Derivative of sin/cos	`d/dx[sin x] = cos x`, `d/dx[cos x] = -sin x`

Strategy

How to study math effectively

Do problems, don't just read them

Watching a worked example feels like understanding, but math is a skill - you only build it by solving problems yourself, including ones you get wrong.

Redo your mistakes from scratch

When you miss a problem, don't just read the correct answer. Close the solution and try the problem again a day later.

Build a formula sheet as you go

Use the Flashcard Builder to turn each new formula into a card the moment you learn it, rather than cramming them all before a test.