Unreasonable equation

Basic Concepts

is apart manner contains unknown equations are irrational equation , General Method irrational equation is the equation has physics and chemistry, into a rational solve the equations.

① transpose square: the square root side toward the rest are on the other side, i.e., the square root is removed, transferred into Zhengshi equation;

② Zhengshi equation solutions;

③ back to the original generation of verification equation, can meet the domain, and vice versa give it.

Note Point: request domain to consider two aspects: the non-negative square root, left and right sides are non-negative after transposition. I.e. as

, after transposition to give

square to root give

, so

, so too

which domain should satisfy

, i.e.,

, only one root of the original equation, i.e. 1.

identify

irrational equation is determined whether an equation of the equation is unreasonable, but only if at the same time meets the definition of the form of equation irrational two conditions: ① containing radical; ② radicand It contains the number of unknowns.

irrational equation to determine whether the real root

Example 1 irrational following equation, there is a real solution of ().

①

②

③

④

⑤

⑥

solution: ① is small problem, the left equation is greater than or equal to 0, and the right side is less than 0 . So no solution.

to ② small problems, can be obtained on both sides of the square root of the equation is x = -2;

The first problem ③ small, equation solutions is irrational in the real number range, it is to secondary radical sense shall

and

can only equal 2, the left side of the equation is equal to 0 and therefore, while the right side is equal to 1, unequal sides, so no solution.

small problem of ④, ③ small problems with the first, to make meaningful radical,

can only be equal to 2, and when

, the left and right sides of the equation We are equal, and thus equation has a solution

⑤ of small problems, according to a non-negative real numbers, can be obtained by

⑥ of small problems, small problems with the first ③. For secondary radical meaningful,

and

i.e.

and

so no solution.

Therefore, there is a real number solution ②④⑤.

Note: a determination method irrational equation no solution mainly through two non-negative real numbers, i.e. radicand (secondary) radical nonnegative (the non-negative), as ⑥; a non-negative value of the second radical (outer non-negative), such as ①, ③ to use the non-negative, but also uses other principles.

unreasonable solution detailed

The basic ideas and the steps unreasonable solution equation equations:

Solution irrational equation, mainly using the " Naturalization of mathematical thought "it into rational equation , the basic method is" both sides of the square ", this step is not the same solution transformation, so must test root . Sometimes a " Conversion method " and other techniques. Conversion method will later be mentioned, observation, etc., can not be separated in fact, the last "on both sides of the square." General Procedure

"on both sides of the square" method

"on both sides of the square" method:

① square sides, into the original equation rational equation;

④ rational solution to this equation,

③ and posterior roots answer: the solution obtained is substituted into the root of the original test irrational equation.

(2) posterior root problem :

Fenshifangcheng different roots and posterior irrational equation. To test not only when it is substituted into the root radical, testing whether a non-negative radicand; but also the whole equation is substituted, it is checked whether the equation for . The following examples of the first example (1) small problem,

substituting radical is meaningful, but substituting into the equation, both sides are not equal, it is by the root.

Example 2 Solutions following irrational equation of the

of:

solution: (1) on both sides of the square, finishing

solve for

Upon examination,

substituting into equation unreasonable, it is by the root of the original equation, rounded down.

Therefore, the root of the original equation is

(2) on both sides of the square, finishing

solving for

Upon examination,

is the original by roots of the equation, rounded down.

Therefore, the root of the original equation is

Conversion method

Example 3 Solutions equation:

Solution: provided

, the original equation may become

(1) when the

so no solution.

(2) when

after test

as a root of the original equation, the roots of the original equation.

This is the second solution solution irrational equation - Conversion method . General Procedure

Exchanging method of solving irrational equations:

(1) observation, analysis of the characteristics equation sought by substitution simple way, set up auxiliary unknown, and containing an auxiliary unknown algebraic equations to represent additional algebraic equations to express additional algebraic expression; new equations for the unknowns of the obtained auxiliary

(2) solution was determined value of the auxiliary unknown;

(3) the auxiliary unknowns are substituted into the original design, the calculated value of the original equation unknowns;

(4) test and answer.

transducer element is typically used when a "on both sides of the square" method or not resolve difficult to resolve (equation is rational polynomial equations), is also commonly used although "on both sides of the square" method can be solved, but more complicated situation.

no matter what method of irrational solutions equation, experimental root is an indispensable step.