x = (-b ± √(b² - 4ac)) / 2a
For a quadratic equation in the form: ax² + bx + c = 0
This formula gives the two solutions (roots) of the equation.
(a + b)ⁿ = ∑(k=0 to n) (n choose k) aⁿ⁻ᵏbᵏ
Expands the power of a binomial into a sum of terms.
Each term is a product of a coefficient, a power of a, and a power of b.
Area: A = πr²
Circumference: C = 2πr
Where r is the radius of the circle and π ≈ 3.14159...
a² + b² = c²
In a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).
V = (4/3)πr³
Where r is the radius of the sphere and π ≈ 3.14159...
a/sin(A) = b/sin(B) = c/sin(C)
In any triangle, the ratio of the length of a side to the sine of the opposite angle is constant.
c² = a² + b² - 2ab·cos(C)
Relates the lengths of the sides of a triangle to the cosine of one of its angles.
sin²(θ) + cos²(θ) = 1
The fundamental trigonometric identity, valid for all values of θ.
Power Rule: d/dx(xⁿ) = n·xⁿ⁻¹
Exponential: d/dx(eˣ) = eˣ
Sine: d/dx(sin(x)) = cos(x)
Cosine: d/dx(cos(x)) = -sin(x)
Power Rule: ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + C, n ≠ -1
Exponential: ∫eˣ dx = eˣ + C
Sine: ∫sin(x) dx = -cos(x) + C
Cosine: ∫cos(x) dx = sin(x) + C
Mean: μ = (∑x) / n
Median: Middle value when ordered
Mode: Most frequent value
σ = √[(∑(x - μ)²) / n]
Measures the amount of variation or dispersion of a set of values.
f(x) = (1 / (σ√(2π))) · e^(-(x-μ)² / (2σ²))
Where μ is the mean and σ is the standard deviation.