Key Mathematical Equations

Mathematical equations are essential tools for solving various problems in math and real life. These notes cover key concepts like the Quadratic Formula, Pythagorean Theorem, and more, helping you understand their applications and significance in different scenarios.

1. Quadratic Formula

   • Used to find the roots of a quadratic equation in the form ax² + bx + c = 0.
   • The formula is x = (-b ± √(b² - 4ac)) / (2a).
   • The discriminant (b² - 4ac) determines the nature of the roots: two real, one real, or two complex.

2. Pythagorean Theorem

   • Relates the lengths of the sides of a right triangle: a² + b² = c², where c is the hypotenuse.
   • Useful for calculating distances and verifying right angles in geometric problems.
   • Can be extended to three dimensions using a² + b² + c² = d².

3. Area of a Circle

   • The area A of a circle is calculated using the formula A = πr², where r is the radius.
   • Important for problems involving circular shapes and real-world applications like land area.
   • The circumference can also be derived from the area: C = 2πr.

4. Slope-Intercept Form

   • The equation of a line in the form y = mx + b, where m is the slope and b is the y-intercept.
   • Allows for easy graphing of linear equations and understanding of line behavior.
   • Slope indicates the steepness and direction of the line.

5. Distance Formula

   • Calculates the distance d between two points (x₁, y₁) and (x₂, y₂) using d = √((x₂ - x₁)² + (y₂ - y₁)²).
   • Essential for problems involving coordinate geometry and real-world distance calculations.
   • Can be derived from the Pythagorean theorem.

6. Exponential Growth/Decay

   • Describes processes that increase or decrease at a rate proportional to their current value, modeled by y = y₀e^(kt).
   • Commonly used in population studies, finance, and natural phenomena.
   • The constant k determines the rate of growth (k > 0) or decay (k < 0).

7. Logarithmic Equations

   • The inverse of exponential functions, expressed as log_b(a) = c, meaning b^c = a.
   • Useful for solving equations involving exponential growth or decay.
   • Properties include the product, quotient, and power rules, which simplify complex logarithmic expressions.

8. Trigonometric Identities

   • Fundamental relationships between trigonometric functions, such as sin²θ + cos²θ = 1.
   • Used to simplify expressions and solve trigonometric equations.
   • Includes reciprocal, quotient, and co-function identities.

9. Derivative Rules

   • Fundamental for finding the rate of change of a function, denoted as f'(x) or dy/dx.
   • Key rules include the power rule, product rule, quotient rule, and chain rule.
   • Derivatives are essential in optimization problems and understanding function behavior.

10. Integration Formulas

    • Used to find the area under a curve or the antiderivative of a function.
    • Fundamental formulas include ∫x^n dx = (x^(n+1))/(n+1) + C for n ≠ -1.
    • Techniques include substitution, integration by parts, and partial fractions for complex functions.