What is an equation?
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Those who take the first steps in algebra, of course,the orderly delivery of material is required. Therefore, in our article that such an equation, we not only give a definition, but also give various classifications of equations with examples.
What is an equation: general concepts
So, the equation is a kind of equality withunknown, denoted by a Latin letter. In this case, the numerical value of this letter, which makes it possible to obtain the correct equality, is called the root of the equation. More about this you can read in our article. What is the root of the equation, we will continue the conversation about the equations themselves. Arguments of the equation (or variables) are called unknowns, and the solution of the equation is the finding of all its roots or the absence of roots.
Types of equations
The equations are divided into two large groups: algebraic and transcendental.
- Algebraic is such an equation, inwhich is used to find the root of the equation only algebraic actions - 4 arithmetic, as well as raising to the power and extracting the natural root.
- An equation is called transcendental, in which non-algebraic functions are used to find the root: for example, trigonometric functions, logarithmic functions, and others.
Among the algebraic equations, we also distinguish:
- whole - with both parts, consisting of whole algebraic expressions in relation to the unknown;
- fractional - containing entire algebraic expressions in the numerator and denominator;
- irrational - algebraic expressions here are under the sign of the root.
We also note that fractional and irrational equations can be reduced to solving entire equations.
Transcendental equations are divided into:
- Indicative - these are the equations thatcontain a variable in the exponent. They are solved by moving to a single base or exponent, putting a common multiplier by the parenthesis, factoring, and some other ways;
- Logarithmic equations with logarithms, thenthere are equations where the unknowns are inside the logarithms themselves. It is very difficult to solve such equations (unlike, say, most algebraic ones), since this requires a solid mathematical preparation. The most important thing here is to go from the equation with the logarithms to the equation without them, that is, to simplify the equation (this way of deleting the logarithms is called potentiation). Of course, it is possible to potentiate the logarithmic equation 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 mastering of trigonometric functions;
- mixed - these are differentiated equations with parts belonging to different types (for example, with parabolic and elliptic parts or elliptic and hyperbolic, etc.).
As for classification by the number of unknowns,then everything is simple: distinguish equations with one, two, three, and so on unknown. There is also another classification, which is based on the degree that exists on the left side of the polynomial. Proceeding from this, linear, square and cubic equations are distinguished. Linear equations can also be called equations of the 1-st degree, square - 2 nd, and cubic, respectively, 3 rd. Well, now we give examples of the 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 0)
Examples of transcendental equations:
- cos x = x lg x = x-5 2x= lgx + x5+40
Examples of integer 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
An 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 in order to solve equationsdifferent types apply a variety of methods. Well, in order to solve almost any equation, knowledge of not only algebra but also trigonometry will be required, and knowledge is often very deep.