Mathematical Formulas for High School

 

High school students are required to master a number of essential mathematical formulas to succeed in their exams. Here is a summary of the main formulas used in algebra, calculus, geometry, and probability.

Algebra

1. Quadratic Equations

   • General form: ( ax^2 + bx + c = 0 )

   • Quadratic formula: ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} )

   • Discriminant: ( \Delta = b^2 - 4ac )

2. Notable Identities

   • ( (a + b)^2 = a^2 + 2ab + b^2 )

   • ( (a - b)^2 = a^2 - 2ab + b^2 )

   • ( a^2 - b^2 = (a + b)(a - b) )

Calculus

1. Derivatives

   • Derivative of ( f(x) = ax^n ): ( f'(x) = nax^{n-1} )

   • Derivative of ( f(x) = e^x ): ( f'(x) = e^x )

   • Derivative of ( f(x) = \ln(x) ): ( f'(x) = \frac{1}{x} )

2. Integrals

   • Integral of ( f(x) = ax^n ): ( \int ax^n , dx = \frac{ax^{n+1}}{n+1} + C ), ( n \neq -1 )

   • Integral of ( f(x) = e^x ): ( \int e^x , dx = e^x + C )

Geometry

1. Trigonometry

   • Fundamental formula: ( \sin^2(x) + \cos^2(x) = 1 )

   • Addition formulas: ( \sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b) ) ( \cos(a \pm b) = \cos(a)\cos(b) \mp \sin(a)\sin(b) )

2. Distance Between Two Points

   • In the plane: ( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} )

Probability

1. Conditional Probabilities

   • ( P(A | B) = \frac{P(A \cap B)}{P(B)} ), if ( P(B) \neq 0 )

2. Expectation and Variance

   • Expectation: ( E(X) = \sum x_i \cdot P(x_i) )

   • Variance: ( V(X) = E(X^2) - [E(X)]^2 )

These formulas cover fundamental concepts that are essential for high school mathematics exams. Mastery of these formulas and their application in various contexts is crucial for success.