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Visual proof of the formulas for the sum and difference of two cubes. In mathematics, the sum of two cubes is a cubed number added to another cubed number.
This means that only one cube root needs to be computed, and leads to the second formula given in § Cardano's formula. The other roots of the equation can be obtained by changing of cube root, or, equivalently, by multiplying the cube root by each of the two primitive cube roots of unity , which are − 1 ± − 3 2 . {\displaystyle {\frac {-1 ...
The solutions of this equation are the x-values of the critical points and are given, using the quadratic formula, by =. The sign of the expression Δ 0 = b 2 – 3ac inside the square root determines the number of critical points. If it is positive, then there are two critical points, one is a local maximum, and the other is a local minimum.
The polynomial x 2 + cx + d, where a + b = c and ab = d, can be factorized into (x + a)(x + b).. In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind.
Sum of four cubes problem, whether every integer is a sum of four cubes; Euler's sum of powers conjecture § k = 3, relating to cubes that can be written as a sum of three positive cubes; Plato's number, an ancient text possibly discussing the equation 3 3 + 4 3 + 5 3 = 6 3
The cube of a number n is denoted n 3, using a superscript 3, [a] for example 2 3 = 8. The cube operation can also be defined for any other mathematical expression, for example (x + 1) 3. The cube is also the number multiplied by its square: n 3 = n × n 2 = n × n × n. The cube function is the function x ↦ x 3 (often denoted y = x 3) that
The general solution formula given above gives the solutions = + +. For any given choice of cube root and its conjugate, this contains nested radicals involving complex numbers, yet it is reducible (even though not obviously so) to one of the solutions 1, 2, or –3.
Completing the cube is a similar technique that allows to transform a cubic polynomial into a cubic polynomial without term of degree two. More precisely, if a x 3 + b x 2 + c x + d {\displaystyle ax^{3}+bx^{2}+cx+d}