(1) Find the root of the original equation, and then substitute it into | x 1 |+x2 | to see if the result is an integer multiple of 2.

(2) Based on the condition that x2-6x-27=0 and x2+6x-27=0 is an even quadratic equation, let c=mb2+n, then we can find its root according to the formula method, and then substitute it into |x 1|+|x2| to get a conclusion.

Solution:

(1) No,

Solve the equation x2+x- 12=0, x 1=3, x2 =-4.

|x 1|+|x2|=3+4=7=2×3.5。

3.5 is not an integer,

∴x2+x- 12=0 is not an even quadratic equation;

(2) existence. The reason for this is the following：

∫x2-6x-27 = 0 and x2+6x-27=0 are even quadratic equations.

∴ suppose c=mb2+n,

When b=-6 and c=-27,

-27=36m+n

X2 = 0 is an even quadratic equation,

∴n=0, three quarters.

∴c=- three quarters? b2。

∵x2+3x？ 27 = 0 is an even quadratic equation,

When b=3, c=- three quarters × 32.

∴ Let c=- 3/4 B2.

For any integer b, c=- three quarters? B2，

△=b2-4ac，

=4b2。

x=

∴x 1=- 3/2 b，x2= 1/2 B

∴|x 1|+|x2|=2|b|，

∫b is an integer,

For any integer b, c=- three quarters? B2, the equation x2+bx+c=0 about x is an even quadratic equation.

Comments: This topic examines the solution of a quadratic equation, the discriminant of roots, the relationship between roots and coefficients, and the application of mathematical modeling ideas. When solving ontology, it is the key to establish a model according to the conditional characteristics.

Ontology test site

Description of the relationship between root and coefficient:

(1) If the coefficient of the quadratic term is 1, the following relations are commonly used: x 1, x2 is two of the equations x2+px+q=0, x 1+x2=-p, x1x2 = q.

(2) If the quadratic coefficient is not 1, the following relationship is commonly used: when x 1, x2 is two of the unary quadratic equation ax2+bx+c=0(a≠0), x 1+x2=, and vice versa.

(3) The relationship between roots and coefficients is often used to solve the following problems:

① Solve the equation and judge whether two numbers are two roots of a quadratic equation; ② Know the equation and one root of the equation, and find the other root and unknown number; ③ Solve the equation and find the value of the formula about the root, such as x 12+x22, etc. ; ④ Judging the symbols of two roots; ⑤ Seeking a new equation; ⑤ Determine the value of letters from the given two satisfying conditions. This kind of problem is more comprehensive.

Solving quadratic equation with one variable by factorization;

Significance of (1) factorization method in solving quadratic equation with one variable

Factorization is a method of solving equations by factorization. This method is simple and easy to use, and it is the most commonly used method to solve the quadratic equation of one variable.

Factorization is to change the right side of the equation into 0 first, and then change the left side into the product of two linear factors through factorization, so that the values of these two factors may be 0, and then we can get the solutions of two linear equations, thus simplifying the original equation and transforming the solution of a quadratic equation into the solution of a linear equation (mathematical transformation idea).

(2) The general steps of factorization to solve a quadratic equation with one variable:

① Move the term so that the right side of the equation is zero; ② decompose the left side of the equation into the product of two linear factors; (3) respectively making each factor zero to obtain two one-dimensional linear equations; (4) Solve these two linear equations, and their solutions are the solutions of the original equation.

Discriminant description of roots:

The discriminant (△=b2-4ac) of the root of a quadratic equation with one variable is used to judge the root of the equation.

The relationship between the root of unary quadratic equation ax2+bx+c=0(a≠0) and △=b2-4ac is as follows:

① When △ > 0, the equation has two unequal real roots;

② When △=0, the equation has two equal real roots;

③ When △ < 0, the equation has no real root.

The above conclusion is also true in turn.