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If the discriminant of such a polynomial is negative, then both roots of the quadratic equation have imaginary parts. In particular, if b and c are real numbers and b 2 − 4 c < 0, all the convergents of this continued fraction "solution" will be real numbers, and they cannot possibly converge to a root of the form u + iv (where v ≠ 0 ...
Multiplying or dividing both sides of an equation by a non-zero quantity. Applying an identity to transform one side of the equation. For example, expanding a product or factoring a sum. For a system: adding to both sides of an equation the corresponding side of another equation, multiplied by the same quantity.
If, on the other hand, the ranks of these two matrices are equal, the system must have at least one solution; since in an underdetermined system this rank is necessarily less than the number of unknowns, there are indeed an infinitude of solutions, with the general solution having k free parameters where k is the difference between the number ...
In general, a system with fewer equations than unknowns has infinitely many solutions, but it may have no solution. Such a system is known as an underdetermined system. In general, a system with the same number of equations and unknowns has a single unique solution. In general, a system with more equations than unknowns has no solution.
The overdetermined case occurs when the system has been overconstrained — that is, when the equations outnumber the unknowns. In contrast, the underdetermined case occurs when the system has been underconstrained — that is, when the number of equations is fewer than the number of unknowns. Such systems usually have an infinite number of ...
Each value of the unknown for which the equation holds is called a solution of the given equation; also stated as satisfying the equation. For example, the equation x 2 − 6 x + 5 = 0 {\displaystyle x^{2}-6x+5=0} has the values x = 1 {\displaystyle x=1} and x = 5 {\displaystyle x=5} as its only solutions.
In both cases, the equations of the theories had unexplained solutions, which led to conjecture of the existence of an unknown particle, and the search for these particles. In both cases, these particles were discovered a few years later by specific experiments.
In geometry, the hinge theorem (sometimes called the open mouth theorem) states that if two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle. [1]