# Beta version

BETA TEST VERSION OF THIS ITEM

This online calculator is currently under heavy development. It may or it may NOT work correctly.

You CAN try to use it. You CAN even get the proper results.

However, please VERIFY all results on your own, as the level of completion of this item is NOT CONFIRMED.

Feel free to send any ideas and comments !

This online calculator is currently under heavy development. It may or it may NOT work correctly.

You CAN try to use it. You CAN even get the proper results.

However, please VERIFY all results on your own, as the level of completion of this item is NOT CONFIRMED.

Feel free to send any ideas and comments !

# Equations by the number of solutions

Equation type | Number of solutions | Description | Example equation | Example solutions |

Contradictory equation | 0 | No solution, there is no such x, that satisfies the equation (our problem can't be solved in such way). | Show source$\sin(x) = 10$ | Show source$-$ |

Equation with one solution | 1 | One solution, there is exactly one x, that satisfies the equation (our problem can be solved in one unique way). | Show source$x + 2 = 5$ | Show source$x = 3$ |

Equation with many solutions | 2, 3, 4... | Many number of solutions, there are more than one x (2,3,4 etc.), that satisfy the equation (our problem can be solved in more than one way). | Show source$x^2 = 4$ | Show source$x = \left\{-2, 2\right\}$ |

Identity equation | ∞ | Infinite number of solutions, no matter what x is, the equation is always satisfied (congratulations ☺, you've just discover another rule, that rules our universe). | Show source$x + x = 2x$ | Show source$x = \left\{-10, 1, \frac{5}{8}, 1000, \dots \right\}$ |

# Equations by the type of unknown

Equation type | Type of unknown (what is x) | Description | Example equation | Example solutions |

Number equation | Number | We search for number, that we can put instead of x to get truth. | Show source$2x + 2 = 5$ | Show source$x = \frac{3}{2}$ |

Functional equation | Function | We search for function, that we can put instead of f to get truth. | Show source$\frac{d}{dx} f(x) = 2x + 1$ | Show source$f(x) = x^2 + x$ |

Diophantine equation | Integer number | We search for integer number, that we can put instead of x to get truth. | Show source$x^2 = 2^x$ | Show source$x = 2$ |

# Number equations by the expression form

Equation type | Expression containing unknown variable(s) | Description | Example equation | Example solutions |

Linear equation (first order) | Show source$a x + b = 0$ | x is unknown value, a and b are known coefficients (x is raised up to the first power). | Show source$2x - 3 = 0$ | Show source$x = \frac{3}{2}$ |

Quadratic equation (second order) | Show source$a x^2 + b x + c = 0$ | x is unknown value, a, b and c are known coefficients (x is raised up to the second power). | Show source$2x^2 - 3x + 1 = 0$ | Show source$x = \left\{1, \frac{1}{2}\right\}$ |

Polynomial equation | Show source$W(x) = a_n x^n + a_{n-1} + a_{n-2} x^{n-2} + \dots + a_1 x + a_0 = 0$ | x is unknown value, a_{n} are known polynomial coefficients (it's a generalization of linear and quadratic equations to polynomial of any range, x can be raised to any power). | Show source$2x^5 + x^4 - 2x-1 = 0$ | Show source$x = \left\{-\frac{1}{2}, -1, -i, i, 1\right\}$ |

Logarithmic equation | Show source$a\ \log(x) - b = 0$ | Unknown value x is located under logarithm function (equation contains logarithm of unknown value). | Show source$\log(x) = 2$ | Show source$x = 100$ |

Exponential equation | Show source$a^x - b = 0$ | Unknown value x is located in exponent (equation contains powering to the unknown number). | Show source$2^x = 8$ | Show source$x = 3$ |

Trigonometric equation | Show source$-$ | Unknown value is located under on of trigonometric function such as sine or tangent (equation contains trigonometric function of unknown argument). | Show source$sin(x) = 1$ | Show source$x = \left\{\dots, -4\pi, -2\pi, 2\pi, 4\pi, \dots\right\}$ |

# Some facts

- In the equation some part is known, and another one is
**unknown**. - Solving the equation consists of finding
**what to substitute**in place of the unknown to get**the true sentence**(equality) . We say then that solution**satisfies the equation**.

ⓘ Example: The solution of the equation:

$2 + x = 5$

is**number 3**because when we replace**x**by it we get:

$2 + 3 = 5$

so:

$5 = 5$ - Equations generally reflect some practical problems in which we
**seek a value that meets our criteria**e.g.:

- we know that 6 eggs are needed for one cheesecake, we want to know
**how many eggs**we need to bake two cheesecakes for the upcoming holidays,

- we need to get to a city 300 km away, we wonder
**how long time**it will take us on a trip if we were going on average at 50 km/h,

- we have pocket money of 10$ per week and we would like to buy a bike for 500$, we wonder
**how many weeks**we have to put off to buy it,

[Li]etc.

- we know that 6 eggs are needed for one cheesecake, we want to know
- Equations can be divided
**due to the type of object**we are looking for (type of solution):

**number equations**- if the solution is the**number**(e.g. number 3), we are looking for answers to questions like:

*"how much"*

*"how many"*,

*"at what price"*,

*"is there a number that..."*,

- etc.

$x = x + 1$**functional equations**- if solution is a**function**(e.g. f(x) = 3x^{2}- 5), we are looking for answers to questions such as:

*"is there a function that ..."*,

*"after which the track will move ..."*,

*"how one value depends on another one"*,

*"the derivative of what function is ..."*,

- etc.

$\dfrac{d}{dx} f(x) = 2x-4$- etc.

- Number equations are often divided due to
**type of expression**that they contain, for example:

**linear equation**- x is an unknown, a and b are known parameters:

$ax + b = 0$**quadratic equation**- x is an unknown, a, b and c are known parameters:

$ax ^ 2 + bx + c = 0$**polynomial equation**- x is an unknown, polynomial coefficients are known:

$W(x) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} \dots a_1 x + a_0 = 0$**exponential equation**- unknown x occurs in the exponent of power:

$a^x - b = 0$**logarithmic equation**- unknown x occurs under the logarithm:

$log(x) - b = 0$- etc.

- Equations can also be divided
**due to the amount of unknowns**:

- equations with one unknown, e.g.

$x = x + 1$ - equation with two unknowns, e.g.

$x + y = 3$ - etc.

- equations with one unknown, e.g.
- Equations can have
**more than one**solution (our problem can be solved in different ways) or**has no solutions**(our problem can't be solved in that way). Depending on the number of solutions, the equation can be divided into:

**identity equation**- there are infinitely many of solutions e.g.

$2(x - 1) + 2 = 2x$(we can find any number of different numbers that when can substituted for x and get the true sentence)

**contradictory equations**- there is no solution, e.g.:

$sin(x) = 10$(there is no such number whose sine is 10)

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