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Learn about the origins and solutions of cubic equations, which are equations of the form ax3 + bx2 + cx + d = 0. Find out how ancient and medieval mathematicians from different cultures approached and solved cubic equations using algebra, geometry, and numerical approximations.
A cube root of a number x is a number y such that y3 = x. Learn how to find cube roots of real and complex numbers, their geometric and algebraic representations, and their applications in mathematics and geometry.
A cubic function is a polynomial function of degree three that maps real or complex numbers to real or complex numbers. Learn how to graph a cubic function, its critical and inflection points, its symmetry, and its classification into three types.
Tetration is an operation based on iterated exponentiation, where n copies of a are raised to a power. Learn how to write and read tetration, how it differs from exponentiation and other hyperoperations, and see some examples of tetration with different bases and heights.
Learn about different algorithms for approximating the non-negative square root of a positive real number, such as Heron's method, Newton's method, and continued fractions. Compare the accuracy, complexity, and history of various methods, and how to choose a suitable initial estimate.
Learn about mental calculation, the arithmetical calculations using only the human brain, with no help from any supplies or devices. Find out the methods and techniques for multiplying, subtracting, and other operations, and see examples and mnemonics.
If the discriminant is zero the fraction converges to the single root of multiplicity two. If the discriminant is positive the equation has two real roots, and the continued fraction converges to the larger (in absolute value) of these. The rate of convergence depends on the absolute value of the ratio between the two roots: the farther that ...
In the case of two nested square roots, the following theorem completely solves the problem of denesting. [2]If a and c are rational numbers and c is not the square of a rational number, there are two rational numbers x and y such that + = if and only if is the square of a rational number d.