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  2. Abel–Ruffini theorem - Wikipedia

    en.wikipedia.org/wiki/Abel–Ruffini_theorem

    In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here, general means that the coefficients of the equation are viewed and manipulated as indeterminates.

  3. Quadratic equation - Wikipedia

    en.wikipedia.org/wiki/Quadratic_equation

    A quadratic equation has at most two solutions. If there is only one solution, one says that it is a double root. If all the coefficients are real numbers, there are either two real solutions, or a single real double root, or two complex solutions that are complex conjugates of each other. A quadratic equation always has two roots, if complex ...

  4. Quadratic formula - Wikipedia

    en.wikipedia.org/wiki/Quadratic_formula

    [29] [30] His solution of the quadratic equation ⁠ + = ⁠ was as follows: "To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is ...

  5. Extraneous and missing solutions - Wikipedia

    en.wikipedia.org/wiki/Extraneous_and_missing...

    Therefore, the solution = is extraneous and not valid, and the original equation has no solution. For this specific example, it could be recognized that (for the value x = − 2 {\displaystyle x=-2} ), the operation of multiplying by ( x − 2 ) ( x + 2 ) {\displaystyle (x-2)(x+2)} would be a multiplication by zero.

  6. Elementary algebra - Wikipedia

    en.wikipedia.org/wiki/Elementary_algebra

    Plot of a quadratic equation (red) and a linear equation (blue) that do not intersect, and consequently for which there is no common solution. In the above example, a solution exists. However, there are also systems of equations which do not have any solution. Such a system is called inconsistent. An obvious example is

  7. Quadratic function - Wikipedia

    en.wikipedia.org/wiki/Quadratic_function

    If a quadratic function is equated with zero, then the result is a quadratic equation. The solutions of a quadratic equation are the zeros (or roots) of the corresponding quadratic function, of which there can be two, one, or zero. The solutions are described by the quadratic formula.

  8. Equation solving - Wikipedia

    en.wikipedia.org/wiki/Equation_solving

    However, if one searches for real solutions, there are two solutions, √ 2 and – √ 2; in other words, the solution set is {√ 2, − √ 2}. When an equation contains several unknowns, and when one has several equations with more unknowns than equations, the solution set is often infinite. In this case, the solutions cannot be listed.

  9. System of polynomial equations - Wikipedia

    en.wikipedia.org/wiki/System_of_polynomial_equations

    Even when the solution set is finite, there is, in general, no closed-form expression of the solutions (in the case of a single equation, this is Abel–Ruffini theorem). The Barth surface, shown in the figure is the geometric representation of the solutions of a polynomial system reduced to a single equation of degree 6 in 3 variables.