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How to solve equations with fractions

How to solve equations with fractions

The equations with fractions - a special kind of equations that has its own specific features and finer points.

Let's try to understand them.



Perhaps the most obvious point here - this,Of course, the denominator. Numeric shot does not pose any danger (fractional equations, where all the denominators are only numbers, will generally be linear), but if the variable is in the denominator, it is necessary to consider and prescribe. Firstly, it means that the value of x is equal to 0 the denominator, the root can not be, and generally need to separately register the fact that X can not be equal to this number. Even if you will, that when substituted in the numerator all perfectly convergent and satisfies. Secondly, we can not multiply or divide both sides by the expression equal to zero.


After that, the solution of this equation is reduced to the transfer of all its members in the left-hand side so that the right was 0.

It is necessary to bring all members to a common denominator, multiplying, where necessary, the numerators for missing expressions.
Next, decide the usual equation, written innumerator. We can make common factors of the brackets, use the formula of abridged multiplication, cause like, to calculate the roots of a quadratic equation in terms of the discriminant, etc.


The result should factorizationas a product of the brackets (x (i-th root)). Also, this may include polynomials without roots, for example, square trinomial with discriminant is less than zero (unless, of course, in the task you want to find only the real roots as often happens).
Necessarily need to factor anddenominator in order to find where the brackets are already contained in the numerator. If the denominator are expressions of the type (x (number)), it is better at bringing to a common denominator standing there brackets are not multiply "head", and leave as a product of the original simple expressions.
Identical brackets in the denominator and numerator can be reduced by writing in advance, as mentioned above, the conditions for x.
The answer is written in braces, as the set of values ​​of x, or simply listing: x1 = ..., x2 = ... etc.

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