The quadratic formula states that for any quadratic equation of the form ax² + bx + c = 0, where a ≠ 0, the solutions are given by x = (-b ± √(b² - 4ac)) / (2a). The expression b² - 4ac is called the discriminant. When the discriminant is positive, the equation has two distinct real roots; when it equals zero, there is exactly one real root (a repeated root); when it is negative, the equation has two complex conjugate roots.


The Pythagorean theorem is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides. This can be written as a² + b² = c², where c represents the length of the hypotenuse and a and b represent the lengths of the triangle's other two sides.


In calculus, the derivative of a function measures the sensitivity to change of the function value with respect to a change in its argument. The derivative of f(x) with respect to x is written as f'(x) or df/dx. For example, the derivative of f(x) = x² is f'(x) = 2x, which means the rate of change of x² at any point x is 2x.