Biquadratic equation

Lecture

ax ⁴ + bx ² + c = 0. - Biquadratic equation.

In order to solve such an equation, it is necessary to make a substitution y = x ² . As a result, we get a quadratic equation. The roots of the original equation are calculated by the formulas:

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Lectures and tutorial on "HANDBOOK ON MATHEMATICS, SCHOOL MATHEMATICS, HIGHER MATHEMATICS"

Terms: HANDBOOK ON MATHEMATICS, SCHOOL MATHEMATICS, HIGHER MATHEMATICS

Math Handbook

Differentiation rules

Derivatives of simple functions, a table of derivatives

Derivatives table, logarithmic and exponential functions

Derivatives table, trigonometric functions

Derivatives table, hyperbolic functions

Integral table, integral of a rational function

The integral table, the integral of the logarithmic function

The integral table, the integral of the exponential function. The improper integral.

Table of integrals, integral of irrational function

Greek alphabet

Latin alphabet

Mathematical constants

Physical constants

Types of algebraic equations

Linear equation definition, examples of solutions

Quadratic equation

Cubic equation

Biquadratic equation

Return (algebraic) equation

Modified and generalized equations of the fourth degree

Generalized return equation

Fourth degree equation of general type

Two-term algebraic equation of nth degree

Special case of equation

Return (algebraic) equation

Algebraic equation of the nth degree of the general form

Types of systems of linear algebraic equations

A system of two linear algebraic equations

A system of m linear algebraic equations

Ordinary differential equations of the first order

Autonomous equation

Equation with separable variables

Linear differential equation

Bernoulli equation

Homogeneous equation

Special type Riccati equation

Special type Riccati equation, case 1

A special type of Riccati equation, case 2 y '= f (x) y2 + ay - ab - b2f (x).

Riccati equation of a special type, case 3 y '= y2 + xf (x) y + f (x).

Special type Riccati equation, case 4 y '= f (x) y2 - axnf (x) y + anxn-1

Special Riccati equation, case 5 y '= f (x) y2 + anxn-1 - a2x2nf (x).

Riccati equation of special type, case 6 y '= - (n + 1) xny2 + xn + 1f (x) y - f (x).

A special type of Riccati equation, case 7 xy '= f (x) y2 + ny + ax2nf (x).

Special Riccati equation, case 8 xy '= x2nf (x) y2 + [axnf (x) - n] y + bf (x).

Riccati equation of a special type, case 9 y '= f (x) y2 + g (x) y - a2f (x) - ag (x).

Special Riccati equation, case 10 y '= f (x) y2 + g (x) y + anxn-1 λ a2x2nf (x) λ axng (x).

A special type of Riccati equation, case 11 y '= aeλxy2 + aeλxf (x) y + λf (x).

Special Riccati equation, case 12 y '= f (x) y2 - aeλxf (x) y + aλeλx.

Riccati equation of special type, case 13 y '= f (x) y2 + aλeλx - a2e2λxf (x).

Special type Riccati equation, case 14 y '= f (x) y2 + λy + ae2λxf (x)

Riccati equation of special type, case 15 y '= y2 - f2 (x) + f' (x).

Special type Riccati equation, case 16 y '= f (x) y2 - f (x) g (x) y + g' (x)

Riccati equation of a special type, the general form y '= f (x) y2 + g (x) y + h (x).

The exponential form of a complex number

Degrees

The roots

Polynomial Horner scheme of the division of P (x) by (x-x0) Bezout theorem

Functions Domain of definition and values Parity and oddness Periodicity Increasing, decreasing function Conversion of graphs of functions

Derivative Rules for calculating derivatives Derivatives table Equation tangent

Newton's formula - Leibniz primitive (indefinite integral). Integral table.

Numeric sets

Names and designations of numerical intervals on the coordinate line

Signs of the divisibility of integers

The law of addition and multiplication of numbers

Real Number Module

Abbreviated Multiplication Formulas

Degrees (m, n are integers)

Square roots (a ≥ 0, b ≥ 0)

Degree with a rational indicator

Proportion

The quadra equation ax² + bx + c (a ≠ 0) and the Vieta theorem

Arithmetic and Geometric Progression