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The roots of the quadratic function y = 1 / 2 x 2 − 3x + 5 / 2 are the places where the graph intersects the x-axis, the values x = 1 and x = 5. They can be found via the quadratic formula. In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation.
This is illustrated by Wilkinson's polynomial: the roots of this polynomial of degree 20 are the 20 first positive integers; changing the last bit of the 32-bit representation of one of its coefficient (equal to –210) produces a polynomial with only 10 real roots and 10 complex roots with imaginary parts larger than 0.6.
A root of degree 2 is called a square root and a root of degree 3, a cube root. Roots of higher degree are referred by using ordinal numbers, as in fourth root, twentieth root, etc. The computation of an n th root is a root extraction. For example, 3 is a square root of 9, since 3 2 = 9, and −3 is also a square root of 9, since (−3) 2 = 9.
has no real number solution since no real number squared equals −1. Sometimes a quadratic equation has a root of multiplicity 2, such as: (+) = For this equation, −1 is a root of multiplicity 2. This means −1 appears twice, since the equation can be rewritten in factored form as
Inverting this process allows square roots to be found, and similarly for the powers 3, 1/3, 2/3, and 3/2. Care must be taken when the base, x, is found in more than one place on its scale. For instance, there are two nines on the A scale; to find the square root of nine, use the first one; the second one gives the square root of 90.
The number of roots of a nonzero polynomial P, counted with their respective multiplicities, cannot exceed the degree of P, [25] and equals this degree if all complex roots are considered (this is a consequence of the fundamental theorem of algebra). The coefficients of a polynomial and its roots are related by Vieta's formulas.
The complex conjugate root theorem states that if the coefficients of a polynomial are real, then the non-real roots appear in pairs of the form (a + ib, a – ib).. It follows that the roots of a polynomial with real coefficients are mirror-symmetric with respect to the real axis.
Analogously, the inverses of tetration are often called the super-root, and the super-logarithm (In fact, all hyperoperations greater than or equal to 3 have analogous inverses); e.g., in the function =, the two inverses are the cube super-root of y and the super-logarithm base y of x.