Equations with fractions are a special kind of equations having its own specific features and subtle moments.
Let's try to understand them.
Perhaps the most obvious moment here is,Of course, the denominator. Numerical fractions do not pose any danger (fractional equations, where only the numbers are in all denominators, will in general be linear), but if the variable is in the denominator, then it is necessary to take into account and prescribe. First, this means that the value of x, which reverses the denominator, can not be a root, and in general it is necessary to separately register the fact that X can not equal this number. Even if you succeed, that when substituted into the numerator everything perfectly converges and satisfies the conditions. Secondly, we can not multiply or divide both sides of the equation by an expression equal to zero.
After this, the solution of such an equation is reduced to the transfer of all its terms to the left side so that 0 remains in the right.
It is necessary to bring all the members to a common denominator, multiplying, where necessary, the numerators for the missing expressions.
Next, we solve the usual equation written inThe numerator. We can put common factors in parentheses, apply formulas of reduced multiplication, give similar ones, calculate the roots of a quadratic equation through a discriminant, etc.
In the end, you should get a factorizationIn the form of a product of brackets (x- (i-th root)). Also, polynomials that do not have roots can enter here, for example, a square trinomial with a discriminant less than zero (if, of course, the problem requires finding only real roots, as often happens).
It is necessary to factorize andDenominator with the purpose of finding there brackets already contained in the numerator. If there are expressions of the type (x- (number)) in the denominator, then it is better, when reducing to a common denominator, the parentheses in it not to be multiplied "in the forehead", but left as a product of the original simple expressions.
The same parentheses in the numerator and denominator can be shortened by writing preliminary, as mentioned above, the conditions on x.
The answer is written in curly brackets, as a set of values of x, or simply by an enumeration: x1 = ..., x2 = ... etc.