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.