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  2. Equation solving - Wikipedia

    en.wikipedia.org/wiki/Equation_solving

    For example, the equation x + y = 2x – 1 is solved for the unknown x by the expression x = y + 1, because substituting y + 1 for x in the equation results in (y + 1) + y = 2(y + 1) – 1, a true statement. It is also possible to take the variable y to be the unknown, and then the equation is solved by y = x – 1.

  3. Solving quadratic equations with continued fractions - Wikipedia

    en.wikipedia.org/wiki/Solving_quadratic...

    By applying the fundamental recurrence formulas we may easily compute the successive convergents of this continued fraction to be 1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, ..., where each successive convergent is formed by taking the numerator plus the denominator of the preceding term as the denominator in the next term, then adding in the ...

  4. Quintic function - Wikipedia

    en.wikipedia.org/wiki/Quintic_function

    where a = ⁠ 5(4ν + 3) / ν 2 + 1 ⁠. Using the negative case of the square root yields, after scaling variables, the first parametrization while the positive case gives the second. The substitution c = ⁠ −m / ℓ 5 ⁠, e = ⁠ 1 / ℓ ⁠ in the Spearman–Williams parameterization allows one to not exclude the special case a = 0 ...

  5. How to Solve It - Wikipedia

    en.wikipedia.org/wiki/How_to_Solve_It

    First, you have to understand the problem. [2] After understanding, make a plan. [3] Carry out the plan. [4] Look back on your work. [5] How could it be better? If this technique fails, Pólya advises: [6] "If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?"

  6. Runge–Kutta–Fehlberg method - Wikipedia

    en.wikipedia.org/wiki/Runge–Kutta–Fehlberg...

    The coefficients found by Fehlberg for Formula 2 (derivation with his parameter α 2 = 3/8) are given in the table below, using array indexing of base 1 instead of base 0 to be compatible with most computer languages:

  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. Risch algorithm - Wikipedia

    en.wikipedia.org/wiki/Risch_Algorithm

    If the constant field is computable, i.e., for elements not dependent on x, then the problem of zero-equivalence is decidable, so the Risch algorithm is a complete algorithm. Examples of computable constant fields are ℚ and ℚ( y ) , i.e., rational numbers and rational functions in y with rational-number coefficients, respectively, where y ...

  9. Quartic function - Wikipedia

    en.wikipedia.org/wiki/Quartic_function

    The four roots of the depressed quartic x 4 + px 2 + qx + r = 0 may also be expressed as the x coordinates of the intersections of the two quadratic equations y 2 + py + qx + r = 0 and y − x 2 = 0 i.e., using the substitution y = x 2 that two quadratics intersect in four points is an instance of Bézout's theorem.