What is an equation?

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What is an equation?

Those who make the first steps in algebra, of course, require the most orderly presentation of the material. Therefore, in our article on what an equation is, we will not only give a definition, but also give various classifications of equations with examples.

What is an equation: general concepts

So, an equation is a kind of equality with an unknown, denoted by a Latin letter. In this case, the numerical value of this letter, which allows to obtain the correct equality, is called the root of the equation. You can read more about this in our article What is the root of the equation, we will continue the conversation about the equations themselves. The arguments of the equation (or variables) are called unknowns, and the solution to the equation is to find all its roots or the absence of roots.

Types of equations

The equations are divided into two large groups: algebraic and transcendental.

  • Algebraic is an equation in which only algebraic actions are used to find the root of the equation - 4 arithmetic, as well as raising to a power and extracting the natural root.
  • Transcendental is an equation in which non-algebraic functions are used to find the root: for example, trigonometric, logarithmic, and others.

Among the algebraic equations there are also:

  • integers - with both parts consisting of integer algebraic expressions with respect to the unknowns;
  • fractional - containing integer algebraic expressions in the numerator and denominator;
  • irrational - algebraic expressions here are under the root sign.

Note also that fractional and irrational equations can be reduced to solving entire equations.

Transcendental equations are divided into:

  • exponential - these are equations that contain a variable in the exponent. They are solved by moving to a single base or exponent, by putting the common factor out of the bracket, by factoring and in some other ways;
  • logarithmic - equations with logarithms, that is, such equations where the unknowns are inside the logarithms themselves. It is not easy to solve such equations (unlike, say, most algebraic ones), since this requires solid mathematical preparation.The most important thing here is to move from an equation with logarithms to an equation without them, that is, to simplify the equation (this method of removing logarithms is called potentiation). Of course, a logarithmic equation can be potentiated only if they have identical numerical bases and do not have coefficients;
  • trigonometric are equations with variables under the signs of trigonometric functions. Their solution requires the initial assimilation of trigonometric functions;
  • mixed are differential equations with parts of different types (for example, with parabolic and elliptic parts or elliptic and hyperbolic, etc.).

As for classification by the number of unknowns, everything is simple: they distinguish equations with one, two, three, and so on unknowns. There is also another classification that is based on the degree that is on the left side of the polynomial. Based on this, linear, quadratic, and cubic equations are distinguished. Linear equations can also be called 1st degree equations, quadratic equations are 2nd, and cubic, respectively, 3rd. Well, now we give examples of equations of a particular group.

Examples of different types of equations

Examples of algebraic equations:

  • ax + b = 0
  • ax3+ bx2+ cx + d = 0
  • ax4+ bx3+ cx2+ bx + a = 0
    (a is not equal to 0)

Examples of transcendental equations:

  • cos x = x lg x = x − 5 2x= lgx + x5+40

Examples of whole equations:

  • (2 + x) 2 = (2 + x) (55x-4) (x2-12x + 10) 4 = (3x + 10) 4 (4x2 + 3x-10) 2 = 9x4

Example of fractional equations:

  • 15 x + - = 5x - 17 x

Example of irrational equations:

  • √2kf (x) = g (x)

Examples of linear equations:

  • 2x + 7 = 0 x - 3 = 2 - 4x 2x + 3 = 5x + 5 - 3x - 2

Examples of quadratic equations:

  • x2+ 5x − 7 = 0 3x2+ 5x − 7 = 0 11x2−7x + 3 = 0

Examples of cubic equations:

  • x3-9x2-46x + 120 = 0 x3- 4x2+ x + 6 = 0

Examples of exponential equations:

  • 5x + 2= 125 3x·2x= 8x + 332x+4·3x-5 = 0

Examples of logarithmic equations:

  • log2x = 3 log3x = -1

Examples of trigonometric equations:

  • 3sin2x + 4sin x cosx + cos2x = 2 sin (5x + π / 4) = ctg (2x-π / 3) sinx + cos2x + tg3x = ctg4x

Examples of mixed equations:

  • logx(log9(4⋅3x−3)) = 1 | 5x − 8 | + | 2⋅5x + 3 | = 13

It remains to add that a variety of methods are used to solve equations of various types. Well, in order to solve almost any equations, you need knowledge not only of algebra, but also trigonometry, and often very deep knowledge.