Algebraic equations

Algebraic equations of the n-th degree, multiple roots, number of roots of an algebraic equation of the n-th degree.

Algebraic functions are functions whose values, for a given value of x, are calculated using only arithmetic operations (addition, subtraction, multiplication and division) and operations of raising to a power (including with a rational exponent). An algebraic equation of degree n with one unknown x is an equation P(x)=0, where P(x) algebraic function of the form P(x)=a0xn + a1xn-1 + ... + an-1x + an, the coefficients ai are real (or complex) numbers, and a0 ≠ 0. The algebraic expression P(x) is called a polynomial of degree n with respect to x, the coefficient an is a free member of the polynomial.
To solve an equation means to find the values ​​of x for which the equality P(x)=0 holds. The found values ​​of x are called the roots of the equation or the zeros of the function P(x).
If the equality P(x)= f(x)(x-x0)m is satisfied, where f(x) is a polynomial and f(x0) ≠ 0,
then the root x=x0 is a root of multiplicity m.
The fundamental theorem of polynomial algebra states that an algebraic equation of degree n has exactly n roots
if a root of multiplicity m is counted m times.
A complete solution of an algebraic equation consists in finding all the roots with their multiplicities.
Only for algebraic equations of the first degree (linear), second degree (quadrate), third degree (cubic) and fourth degree (quartic),
there are formulas expressing the roots of these equations through their coefficients using a finite number of arithmetic operations
(addition, subtraction, division, multiplication) and root extractions. There are no such formulas for equations of higher degrees.


Linear equations

Quadratic equations, biquadratic equations

            Solving of incomplete quadratic equations

             Solving of complete quadratic equations

             Vieta's theorem and its application to solving quadratic equations

            Solving of biquadratic equations

Cubic equations

            Solving of cubic equations by factorization

             Solving of symmetric cubic equations

            Cardano's formula for solving cubic equations

Equations of the fourth degree

             Solving of quartic equations by factorization

            Solving of symmetric quartic equations

             Ferrari's method for solving quartic equations

Equations of higher degrees

Fractional-rational equations

Equations with modulus

Irrational equations

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