There are several solutions to the unary quadratic equation: the first one is factorization, and the factorization methods are: (1) cross multiplication (including the quadratic coefficient of 1 and the quadratic coefficient of 1, but excluding 0), and (2) formula method: (including the complete square formula and the standard deviation formula. ). (3) Example of extracting common factor formula 1: x 2-4x+3 = 0 In this question, the original equation is decomposed into (X-3)(X- 1)=0 by cross multiplication in factorization, and X=3 or 1 can be obtained. Example 2: x 2-8x+ 16 = 0 This problem uses the complete square formula in the factorization method to decompose the original equation into (x-4) 2 = 0, thus obtaining X 1=4 X2=4 (Note: when encountering such problems, it must be written as x1= x2. Example 3: x 2-9 = 0 This question uses the square difference formula in factorization. The original equation is decomposed into (X-3)(X+3)=0, and X 1=3 and X2=-3 can be obtained. Example 4: The problem of x 2-5x = 0 is solved by extracting the common factor method in factorization. When the original equation is decomposed into X(X-5)=0, we can get X 1=0 and X2=5. The second method is collocation, which is more complicated. The following examples illustrate how to solve a quadratic equation with one variable by collocation method: x 2+. Step 2: The original formula is x 2+2x-3, (x+ 1) 2 = x 2+2x+ 1. After comparing two sunflower seeds, it is found that the original formula will be equal only if the constant term 4 is subtracted, so the formula obtained by matching method is (x+65438+). Another method is Kaiping method, for example: x 2 = 12 1, then x1=1,x2 =- 1 1 Finally, if the above methods can't solve the equation, we can only use the formula for finding the root mentioned above. Theorem is the discriminant of Vieta theorem and root. Vieta's theorem is the unary equation AX 2+BX+C = 0 (A is not equal to 0). The sum of two roots is -b/a, and the product of two roots is c/a. For example, the sum of two roots is -(-4/ 1)=4. Factorization method: decompose the quadratic trinomial on one side of the equation into the product of two linear factors, so that the two linear factors are equal to zero respectively, and two linear equations are obtained. The roots obtained by solving these two linear equations are the two roots of the original equation. This method of solving a quadratic equation with one variable is called factorization. Example 4. Solve the following equations by factorization: (1) (x+3) (x-6) =-8 (2) 2x2+3x = 0 (3) 6x2+5x-50 = 0 (optional research) (4) x2-2 (+) x+4. =-8 Simplify the arrangement to get x2-3x- 10=0 (quadratic trinomial on the left and 0 on the right) (x-5) (x+2) = 0 (factorization on the left side of the equation) ∴x-5=0 or x+2=0 (converted into two linear equations) \ It should be remembered that there are two solutions to the quadratic equation of one variable. (3) Solution: 6x2+5x-50 = 0 (2x-5) (3x+10) = 0 (pay special attention to the symbols when factorizing cross multiplication) ∴2x-5=0 or 3x+/kloc-0 = 0 ∴ x656;. (4) Solution: x2-2(+ )x+4 =0 (∵4 can be decomposed into 2? 2. ∴ This problem can be factorized) (x-2)(x-2 )=0 ∴x 1=2, and x2=2 is the solution of the original equation. Summary: Generally speaking, factorization is the most commonly used method to solve quadratic equations with one variable. When factorization is applied, the equation is written in a general form and the quadratic coefficient is turned into a positive number. Direct leveling method is the most basic method. Formula and collocation are the most important methods. Formula method is suitable for any quadratic equation with one variable (some people call it universal method). When using the formula method, the original equation must be transformed into a general form to determine the coefficient, and before using the formula, the value of the discriminant should be calculated to judge whether the equation has a solution. Matching method is a tool to derive formulas. After mastering the formula method, we can directly use the formula method to solve the quadratic equation of one variable, so we generally don't need to use the matching method to solve the quadratic equation of one variable. However, collocation method is widely used in the study of other mathematical knowledge, and it is one of the three important mathematical methods required to be mastered in junior high school, so we must master it well. Three important mathematical methods: method of substitution, collocation method and undetermined coefficient method.