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  2. Equation solving - Wikipedia

    en.wikipedia.org/wiki/Equation_solving

    In some other cases, in particular if the equation is in one unknown, it is possible to solve the equation for rational-valued unknowns (see Rational root theorem), and then find solutions to the Diophantine equation by restricting the solution set to integer-valued solutions. For example, the polynomial equation

  3. Cubic equation - Wikipedia

    en.wikipedia.org/wiki/Cubic_equation

    A cubic equation with real coefficients can be solved geometrically using compass, straightedge, and an angle trisector if and only if it has three real roots. [30]: Thm. 1 A cubic equation can be solved by compass-and-straightedge construction (without trisector) if and only if it has a rational root.

  4. Rational root theorem - Wikipedia

    en.wikipedia.org/wiki/Rational_root_theorem

    If the rational root test finds no rational solutions, then the only way to express the solutions algebraically uses cube roots. But if the test finds a rational solution r, then factoring out (x – r) leaves a quadratic polynomial whose two roots, found with the quadratic formula, are the remaining two roots of the cubic, avoiding cube roots.

  5. Recurrence relation - Wikipedia

    en.wikipedia.org/wiki/Recurrence_relation

    See Rational difference equation and Matrix difference equation for example of uses of "difference equation" instead of "recurrence relation" Difference equations resemble differential equations, and this resemblance is often used to mimic methods for solving differentiable equations to apply to solving difference equations, and therefore ...

  6. Diophantine equation - Wikipedia

    en.wikipedia.org/wiki/Diophantine_equation

    Solving a homogeneous Diophantine equation is generally a very difficult problem, even in the simplest non-trivial case of three indeterminates (in the case of two indeterminates the problem is equivalent with testing if a rational number is the d th power of another rational number).

  7. Algebraic equation - Wikipedia

    en.wikipedia.org/wiki/Algebraic_equation

    If an equation P(x) = 0 of degree n has a rational root α, the associated polynomial can be factored to give the form P(X) = (X – α)Q(X) (by dividing P(X) by X – α or by writing P(X) – P(α) as a linear combination of terms of the form X k – α k, and factoring out X – α. Solving P(x) = 0 thus reduces to solving the degree n – 1 ...

  8. Fermat's Last Theorem - Wikipedia

    en.wikipedia.org/wiki/Fermat's_Last_Theorem

    Fermat's Last Theorem considers solutions to the Fermat equation: a n + b n = c n with positive integers a, b, and c and an integer n greater than 2. There are several generalizations of the Fermat equation to more general equations that allow the exponent n to be a negative integer or rational, or to consider three different exponents.

  9. Nested radical - Wikipedia

    en.wikipedia.org/wiki/Nested_radical

    This rational number can be found by realizing that x also appears under the radical sign, which gives the equation x = 2 + x . {\displaystyle x={\sqrt {2+x}}.} If we solve this equation, we find that x = 2 (the second solution x = −1 doesn't apply, under the convention that the positive square root is meant).