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Even for the first root that involves at most two square roots, the expression of the solutions in terms of radicals is usually highly complicated. However, when no square root is needed, the form of the first solution may be rather simple, as for the equation x 5 − 5 x 4 + 30 x 3 − 50 x 2 + 55 x − 21 = 0 , for which the only real solution is
Polynomial equations of degree two can be solved with the quadratic formula, which has been known since antiquity. Similarly the cubic formula for degree three, and the quartic formula for degree four, were found during the 16th century. At that time a fundamental problem was whether equations of higher degree could be solved in a similar way.
A solution in radicals or algebraic solution is an expression of a solution of a polynomial equation that is algebraic, that is, relies only on addition, subtraction, multiplication, division, raising to integer powers, and extraction of n th roots (square roots, cube roots, etc.). A well-known example is the quadratic formula
In this vein, the discriminant is a symmetric function in the roots that reflects properties of the roots – it is zero if and only if the polynomial has a multiple root, and for quadratic and cubic polynomials it is positive if and only if all roots are real and distinct, and negative if and only if there is a pair of distinct complex ...
A method analogous to piece-wise linear approximation but using only arithmetic instead of algebraic equations, uses the multiplication tables in reverse: the square root of a number between 1 and 100 is between 1 and 10, so if we know 25 is a perfect square (5 × 5), and 36 is a perfect square (6 × 6), then the square root of a number greater than or equal to 25 but less than 36, begins with ...
For polynomials with real or complex coefficients, it is not possible to express a lower bound of the root separation in terms of the degree and the absolute values of the coefficients only, because a small change on a single coefficient transforms a polynomial with multiple roots into a square-free polynomial with a small root separation, and ...
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.
The 2nd-order super-root, square super-root, or super square root has two equivalent notations, () and . It is the inverse of 2 x = x x {\displaystyle ^{2}x=x^{x}} and can be represented with the Lambert W function : [ 18 ]