1.=2.……( )
2. This is a quadratic radical ............... ()
3.== 13- 12= 1.( )
4.,, is a quadratic radical of the same kind ... ()
5. The physical and chemical factors are ........... ().
Answer 1. √; 2.×; 3.×; 4.√; 5.×.
(2) Fill in the blanks:
6. Equation = 1-x holds if _ _ _ _ _ _ _.
The answer x ≤ 1.
7. When x _ _ _ _ _ _ _ _ _ _ the quadratic radical is meaningful.
The answer is ≥
8. Comparison size:-2 _ _ _ _ 2-.
answer
9. Calculation: equal to _ _ _ _ _ _.
Answer 2.
10. Calculation: = _ _ _ _ _ _ _ _ _ _.
The answer.
1 1. The positions of the corresponding points of real numbers A and B on the number axis are as follows: a o b is 3a-= _ _ _ _ _ _ _ _ _ _.
Answers 6a-4b.
12. If += 0, then X = _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.
Answer 8, 2.
The physical and chemical factors of 13.3-2 are _ _ _ _ _ _.
Answer 3+2.
14. When < x < 1,-= _ _ _ _ _ _.
Answer-2x ..
15. If the simplest quadratic radical is the same quadratic radical, then a = _ _ _ _ _ _ _ _ _ _ _ _ and b = _ _ _ _ _ _.
Answer 1, 1.
(3) multiple choice questions:
16. Among the following variants, the correct one is ............ ().
(A)(2)2=2×3=6 (B)=-
(C)= (D)
Answer D.
Comment on this topic to examine the properties of quadratic roots. Note (b) is incorrect because = |-| =; (c) Incorrect because there is no formula =.
17. Among the following categories, the category that must be correct is ... ()
(A)=a+b (B)=a2+ 1
(C)= (D)
Answer B.
Comment on this topic to investigate the conditions for the establishment of quadratic radical properties. (a) is incorrect, because a+b is not necessarily negative, (c) must be a≥ 1, and (d) must be a≥0 and b > 0.
18. If the formula -+ 1 is meaningful, the value range of x is ........................... ().
(a) x ≥ (b) x ≤ (c) x = (d) or more is incorrect.
Tip To make a formula meaningful, you must
Answer C.
19. When a < 0, b < 0, it is reduced to the simplest quadratic root, and …………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………….
(A) (B)- (C)- (D)
Tip = =。
Answer B.
Comment on this question. Nature = | a | and denominator are rational numbers. Note (a) The reason for the error is that the number is not considered when using nature.
20. When a < 0, the result of simplifying | 2a-| is ... ()
(A)a (B)-a (C)3a (D)-3a
Prompt to simplify first, ∫a < 0, ∴ =-a. Then simplify | 2a-| = | 3a |.
Answer D.
(4) Factorization in the range of real numbers:
2 1.2x 2-4；
It is suggested to extract 2 first, and then use the square difference formula.
Answer 2 (x+) (x-).
22.x4-2x2-3。
It is suggested that x2 should be regarded as a whole first, and then decomposed by x2+px+q = (x+a) (x+b), where a+b = p and ab = q, and then decomposed by the square difference formula.
Answer (x2+ 1) (x+) (x-).
(5) Calculation:
23.(-)-(-);
It is suggested that each quadratic root should be transformed into the simplest quadratic root first, and then similar quadratic roots should be merged.
The answer.
24.(5+-)÷;
Solution formula = (20+2-) × = 20×+2×-
=20+2-×=22-2.
25.+-4+2(- 1)0;
The solution formula is = 5+2 (- 1)-4×+2× 1.
=5+2-2-2+2=5.
26.(-+2+)÷.
It is suggested that division be converted into multiplication, multiplied by the distribution law and then simplified.
Solve the original formula = (-+2+)
= - +2 +
=-+2+=a2+a-+2。
It is complicated to simplify the items in brackets and then multiply them by the distribution law.
(6) Evaluation
27. Given a =, b =, find the value of-.
It is suggested to simplify the quadratic root first and then substitute it for evaluation.
Solve the original formula = = =.
When a = and b =, the original formula = = 2.
Comments If the values of A and B are directly substituted into the calculation, the calculation process is complicated and calculation errors are easy to occur.
28. Given x =, find the value of x2-x+.
It is suggested that this question be simplified after X, and then substituted for evaluation.
Solution ∵ x = =.
∴x2-x+=(+2)2-(+2)+= 5+4+4-2+= 7+4。
If we can notice that X-2 = thus (X-2) 2 = 5, we can also change X-2 =-X+ into about.
The quadratic trinomial of x-2 can be solved as follows:
∫x2-x+=(x-2)2+3(x-2)+2+=()2+3+2+= 7+4。
Obviously, the operation is convenient, but the constant deformation of the formula is very demanding.
29. Given += 0, find the value of (x+y) x 。
Hints are all arithmetic square roots, so they are all non-negative. What is the conclusion that the sum of two non-negative numbers is equal to 0?
Solution: ≥0, ≥0,
And += 0,
∴ (x+y) x = (2+ 1) 2 = 9。
(7) solving problems:
30. It is known that the hypotenuse length of a right-angled triangle is (2+) cm and the right-angled side length is (+2+)cm. Find the area of this right-angled triangle.
What do you need to find the area of a right triangle in this question? [Another right-angled edge. ] How did you find it? 【 Using Pythagorean Theorem. ]
The solution is in a right triangle. According to Pythagorean theorem:
The length of the other right angle = 3 (cm).
The area of a right triangle is:
S=×3×()=(cm2)
Answer: The area of this right triangle is () square centimeters. ..
3 1. Given | 1-x |-= 2x-5, find the range of x. 。
The hint is given by the known | 1-x |-x-4 | = 2x-5. When was this formula established? [1-x ≤ 0 and x-4 ≤ 0. ]
Since the solution is known, the left side of the equation = |1-x |-=|1-x |-x-4, and the right side = 2x-5.
Only when | 1-x | = x- 1, | x-4 | = 4-x, left side = right side. At this point, the solution is 1 ≤ x ≤ 4. The value range of ∴ x is 1 ≤ x ≤ 4.