# Уравнения

## Содержание

* [#opredelenie](#opredelenie "mention")
* [#svoistva](#svoistva "mention")
* [#ravenstvo-funkcii](#ravenstvo-funkcii "mention")
* [#stepennye](#stepennye "mention")
* [#proizvedenie](#proizvedenie "mention")
* [#irracionalnye](#irracionalnye "mention")
* [#summa-neotricatelnykh-slagaemykh](#summa-neotricatelnykh-slagaemykh "mention")

## Определение

Уравнение — выражение вида $$a=b$$, где $$a,b$$ — числовые выражения.

Область допустимых значений переменных в уравнении $$D(a)\cap D(b)$$.

## Свойства

1. $$a=b \iff a+c=b+c$$,  при $$D(a) \cap D(b)=D(a) \cap D(b) \cap D(c)$$.
2. $$a=b \iff a \cdot c =b \cdot c$$,  при $$c \ne 0,\~D(a) \cap D(b)=D(a) \cap D(b) \cap D(c)$$.

## Равенство функций

$$
\large
f(a)=f(b) \iff \begin{cases} a=b \ a\in D(f) \end{cases}
$$

$$f$$ — монотонная функция.

## Степенные

1. $$a\_nx^n+\ldots+a\_2x^2+a\_1x+a\_0=0 \iff a\cdot b=0 \iff \left\[ \begin{array}{l} a=0\b=0 \end{array} \right.$$
2. $$a^n=b^n \iff  \left\[ \begin{array}{l} a=b\a=-b \end{array} \right.$$,  при чётном $$n>0$$
3. $$ax^2+bx+c=0 \iff \left\[ \begin{array}{l} x=x\_1 \ x=x\_2 \end{array} \right.$$,  при $$a\ne 0,\~D\geqslant 0$$

## Произведение $$=0$$

$$
\large
\dfrac{a\_1 \cdot a\_2 \cdot\ldots\cdot a\_n} {b\_1 \cdot b\_2 \cdot\ldots\cdot b\_m}=0 \iff \begin{cases}
\left\[\begin{array}{l}
a\_1=0 \\
a\_2 = 0 \\
\dots \\
a\_n = 0
\end{array}\right. \\
\mathrm{\mathbf{x}} \in D(a\_1) \\
\mathrm{\mathbf{x}} \in D(a\_2) \\
\dots \\
\mathrm{\mathbf{x}} \in D(a\_n) \\
b\_1\ne 0 \\
b\_2\ne 0\\
\dots \\
b\_m\ne 0
\end{cases}
$$

1. $$a\cdot b=0 \iff \left\[ \begin{array}{l} \begin{cases} a=0 \ b\in D(a) \end{cases} \ \begin{cases} b=0 \ a\in D(b) \end{cases}  \end{array} \right.$$
2. $$\dfrac{a}{b}=0 \iff  \begin{cases} a=0\b\ne 0 \end{cases}$$

## Иррациональные

$$
\large \sqrt\[n]{a}=b \iff \begin{cases} a=b^n\b \geqslant 0 \end{cases}
$$

$$n$$ — чётное число.

## Сумма неотрицательных слагаемых

$$
\large a\_1+a\_2 + \ldots + a\_n=0 \iff \begin{cases} a\_1 = 0 \ a\_2 = 0\ \ldots \ a\_n=0 \end{cases}
$$

все слагаемые $$a\_i \geqslant 0$$.


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