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Sorting out the knowledge points in the first two chapters of the first volume of eighth grade mathematics
Chapter 12 of the first volume of eighth grade mathematics

4. It is known that both p and q are prime numbers, and the tangent satisfies 5P2 +3Q=59. What is a triangle with P+3, 1-P+Q and 2P+Q-4 as sides?

5. As shown in the figure, three angular bisectors in △ABC intersect at point O, and it is known that AB < BC < CA, and verified that OC > OA > OB.

6. Fold a wire with a length of 2n(n is a natural number, n≥4) into a triangle, and the length of each side is an integer. Note that (a, b, c) is a triangle whose three sides are A, B, C and satisfy A < B < C. For the case of n=6, write all (a, b, c) that satisfy the meaning of the question respectively.

7. As shown in the figure, in RT△ABC, D is the midpoint of AC, DE⊥AB and E. Verification: BE2-AE2=BC2.

real number

First of all, mind mapping

1. Definition of irrational number: infinite acyclic decimal

2. Classification of real numbers: divided into rational numbers and irrational numbers. Rational numbers are divided into positive rational numbers, negative rational numbers and zero.

3. Arithmetic square root: If the square of a positive number X is equal to A, that is, x=a, then this positive number X is the arithmetic square root of A, and the arithmetic square root of A is recorded as "root number A", and A is called root number. Rule: The arithmetic square root of 0 is 0.

4. square root: if the square of a number x is equal to a, that is, x=a, then this number x is called the square root of a.

5. Definition of quadratic root: An algebraic formula with the general form (a≥0) is called quadratic root, where a is called root number, and the root number must be greater than or equal to 0.

6. The simplest quadratic root satisfaction: ①. Denominator does not contain root sign = there is no denominator under root sign = there is no fraction under root sign.

(2) The radical sign does not contain numbers that can be completely opened.

7. Similar quadratic roots: after several quadratic roots are transformed into the simplest quadratic roots, if the number of roots is the same, these quadratic roots are called similar quadratic roots.

8.()2=a (a≥0)? =a(a≥0)

① multiplication rule of quadratic root: × (a≥0, b≥0)

Multiply two quadratic roots, multiply by the root number, and the root index remains unchanged.

② The nature of the arithmetic square root of the product: (a≥0, b≥0)

The arithmetic square root of the product of two nonnegative numbers is equal to the arithmetic square root of these two factors.

③ Division rule of quadratic root: = (a≥0, B > 0)

Divided by two quadratic roots, divided by the root number, the root index remains unchanged.

④ the nature of the arithmetic square root of quotient: = (a≥0, b > 0)

Second, it is easy to make mistakes.

1. Known: = x-+2, find-.

Solution: ∫x-2≥0, 2-x ≥ 0.

∴x=2,= ×2-0+0= 1

Substitute x=2, = 1 into the formula to obtain

Original formula = =3-3=0

2. The following statement: ① Only positive numbers have square roots; ②-2 is the square root of 4; The square root of ③5 is; ④ Both of them are the square roots of 3; The square root of ⑤ is -2, where the correct one is ().

A.①②③ B.③④⑤ C.③④ D.②④

Solution: Error Cause ①: The square root of 0 is 0.

The square root of 3: 5 is positive or negative.

⑤: The square root of is 2 (any non-negative square root is non-negative).

So choose d

3. If the sum is reciprocal, find the value of.

Solution: ∵ ≥0, ≥0.

Again, they are opposites.

∴ = =0

That is, a-b+2=0 b=

A+b- 1=0 gives a=-

Substitute in the original formula and you get

Original formula = =-2

A: The value of the formula is -2.

4. Known 0

Solution: The original formula can be simplified as follows

∵0 1

∴x-<; 0

∴ Original formula = x++x-= 2x

5. Simplify first, then evaluate. -where x=4, =27.

Solution: Original formula =6

=-

6. It is known that the square root of 2+ 1 is 3 and the arithmetic square root of 2. Find the square root of +2n.

Solution: You understand the meaning of the topic.

2+ 1=

=

Solution, =4, n= 18

∴+2n=40

So the square root of +2n is.

7. The range of+meaningful X is ()

a . x≥0 b . x≠2 c . x & gt; Two-dimensional x ≥ 0 and x≠2

Solution: the value range of meaningful x is x≥0,

The range of meaningful x is x-2≠0, x-2 >; 0.

To sum up, the value range of x that makes+meaningful is x >;; 2.

8, known, and, find the value of x+.

Solution: ∵ ≥0, ≥0

It's also VIII

∴ =2, = 1

∵ Again, that is, x-≤0.

Or.

X+=- 1 or 2

9, the following calculation is correct ()

A、

B,

C,

d 、( x & gt0,≥0)

Solution: Error: A. It should be B. It should be C. It should be D.

10, is there a positive integer a and b(a

Solution: existence.

Because only similar quadratic roots can be merged, they are similar quadratic roots.

set up

So +n=6, and a, b, a.

solve

=

that is

=

Available.

Third, think about the problem

1. Let x be a positive rational number and 0 be an irrational number. Prove that+is irrational.

2. Let X and+be integers, and prove that they are integers.

3. If the real number X satisfies 3+5 ~ = 7, find the value range of S = 2-3 ~.

4. There are the following three propositions:

(a) If A and B are unequal irrational numbers, ab+a-b is irrational.

(b) If A and B are unequal irrational numbers, they are irrational numbers.

(c) If A and B are unequal irrational numbers, then+is irrational.

The number of correct propositions is ()

(A)0 (B) 1 (C)2 (D)3

5.2 =

calculate

calculate

8. Given that the integer x satisfies, then the number of integer pairs (x,) is

9. It is known that A, B and C are positive integers and rational numbers, and it is proved that they are integers.

10. Given the real number x, it satisfies (,verification: x+=0.