Analysis: replace the letters in the algebraic expression with letter values, and calculate the results according to the operation specified in the algebraic expression.
When a = 3 and b =,
The original formula = (-3) 2+(-3) ×+3× () 2 = 9-2+3× =.
Example 2: When x=, y=- and y =-, find the value of the algebraic expression (5x-3y)-(2x-y)+(3x-2x).
Analysis: direct substitution, too many items, large amount of calculation; It's easier if you simplify it first and then substitute it.
Original formula = 5x-3y-2x+y+3y-2x = x+y,
When x = and y =-, the original formula =-
Example 3: Given a2-a-4=0, find the value of A2-2 (A2-a+3)-(A2-a-4)-a. 。
Analysis: Carefully observe the known formulas, all of which contain a2-a. You can convert a2-a-4=0 into a2-a=4, and then directly substitute the value of a2-a into the formula.
a2-2(a2-a+3)-(a2-a-4)-a = a2-a-2(a2-a+3)-(a2-a-4)
=(a2-a)-2(a2-a)-6-(a2-a)+2 =-(a2-a)-4。
So when a2-a=4, the original formula =-× 4-4 =- 10.
Example 4. Given, find the value of algebraic expression.
Analysis: the value that can't be obtained directly from the known conditions can't be obtained by =7 and solving the equations, so we should consider how to construct the algebraic formula into a formula containing sum through deformation and then substitute it as a whole.
= =2
∵ ,
The original formula = 2 (7+ 19) = 52.
Example 5, it is known that-1 < b < 0, 0 < a < 1, then in the algebraic expressions A-B, a+b, a+b2, a2+b, for any A and B, the maximum value of the corresponding algebraic expression is
(A) a+b (B) a-b (C) a+b2 (D) a2+b
Analysis: take,, and substitute into four selected branches for calculation: (a) 0; The value of (b) is1; The value of (c) is; The value of (d) is.
Select "b"
Example 6, Setting Rules
Analysis: It happens to be the current value. So, the left is 0 and the right is 0, so =0.
Fill in 0.
Example 7: It is known that A and B are reciprocal, C and D are reciprocal, and the absolute value of X is equal to 1. Find the value of algebraic formula A+B+X2-CDX.
Analysis: according to the meaning of the question, it is: A+B = 0, CD = 1, X = 1.
When x= 1, the original formula = 0+12-1×1= 0.
When x =- 1, the original formula = 0+(-1) 2-1× (-1) = 2.
So the algebraic value is 2 or 0.
Example 8: Given m2-m- 1=0, find the value of the algebraic formula m3-2m+2005.
Analysis: Because m3 = m? M2, m2 = m+ 1, which can be substituted for order reduction.
Because m2-m- 1=0, m2-m = 1, m2 = m+ 1.
So m3 = m? m2=m(m+ 1)=m2+m。
So the original formula = m2+m-2m+2005 = m2-m+2005 =1+2005 = 2006.
Example 9: Given A = 2b and C = 3a, find the value of A2+32b2-C2+3.
Analysis: Take B as known, use B for C, reduce it to the same letter with the idea of reduction, and then substitute it for evaluation.
Because a = 2b, c = 3a and c=6b.
Replaced:
The original formula = (2b) 2+32b2-(6b) 2+3 = 4b2+32b2-36b2+3 = 3.
Simplify it first, and then replace a with an appropriate number: a-3a+1-a+2a-3a? -25 cents? -6a+9
a - a/3 + 1 - a + a/2 - 3/a？ - a？ /4 - 6a + 9 = -35a/6 - 3/a？ - a？ /4 + 10
A takes 6, which means-35-112-9+10 =-34 and112.
Simplify (a+1/a-1/a-2a+1) by a/a- 1, and then select a value a that makes the original formula meaningful for evaluation.
(A+ 1/A- 1/A-2A+ 1)/(A- 1)
=[(a？ - 1)/(a- 1)？ + 1/(a- 1)？ ]*(a- 1)/a
=[a？ /(a- 1)？ )]*(a- 1)/a
If x = 2, example 1
Then x+ 1 = 2+ 1 = 3.
If x = 8
Then x+4 = 8+4 = 12.
If x = 6, example 1
X- 1 = 6- 1 = 5。
Example 2 if x = 7
Then x-3 = 7-3 = 4.
Example 4 If the value of 2x+2002 is equal to _ _ _ _. (2002 Hope Cup National Mathematics Invitational Tournament)
Thinking and analysis can directly calculate the value of x, which students can't find now, but if they can find it, the calculation will be very complicated, so we must find a better way to solve it.
200 1=200 1
Example 3 If x:y:z=3:4:7, 2x-y+z= 18, what is the value of x+2y-z?
Think and analyze the form that x:y:z=3:4:7 can be written. For equal ratio, we can usually set their ratio as a constant k, which will bring convenience to the solution of the problem.
Let x = 3k, y = 4k and z = 7k.
Because 2x-y+z= 18, 2x3k-4k+7k = 18,
So k=2, so x=6, y=8, z= 14,
So x+2y-z = 6+ 16- 14 = 8.
Example 6 If ab= 1, the value of.
There are many solutions to this problem. The key is how to make full use of ab= 1, such as ab= 1, and then directly substitute it into the calculation. For example, replace "1" in the formula with ab= 1; For example, multiply a or b by the numerator and denominator of a fraction in the formula, and then convert it into the same denominator for calculation.
The solution 1 is obtained by ab= 1, so =
Solution 2 ∵ab= 1,∴ =
Solution 3 ∵ab= 1,∴ =
Comments: The substitution of "1" is skillfully used in solution 2 and solution 3 of this problem, and the substitution of "1" is one of the commonly used methods in identity deformation.
That's all I can find. Hey, hey, sorry.