First, the multiplication formula method

Example 1 calculation:

Analysis: Because 2=, the common factor can be extracted in.

Solution: Original formula =

=××

= 19

Second, the factorization method

Example 2 Simplification:

Analysis: The conventional practice of this problem is to make the denominator physical and chemical first, and then calculate it. Unfortunately, the amount of calculation is too large to take. But we found that (x-y) and (x+y-) can be factorized in the real number range, so we have the following methods.

Solution: Original formula =

=

=0.

Third, the whole replacement method.

Example 3 Simplification.

Analysis: The two fractions of this algebraic expression are reciprocal, and the amount of direct calculation is quite large. Might as well find another way, let =a, =b, then a+b=2, ab= 1.

Solution: Original formula =

=

=

=

=4x+2

Fourth, the ingenious constant replacement method

Example 4 is well known.

Analysis: Given the condition in the form of (x0), the items contained in the formula can be converted into = first, that is, a constant is constructed first, and then substituted for evaluation.

Solution: Obviously x0 becomes =3.

The original formula ===2.

quadratic radical

I. Definition:

A formula in the form of √ā(a≥0) is called a quadratic radical.

Two. The range of quadratic root √ ā

√āis a non-negative number. That is, √ā≥0.

When a > 0, √ ā represents the arithmetic square root of a.

When a=0, √ ā represents the arithmetic square root of 0, that is, 0.

Three. Calculation formula:

1.(√ā)? =a(a≥0)

2. when a > 0, √ ā? =a

When a=0, √ ā? =0

When a < 0, √ ā? =-a

3.√ā×√\\\\\\( a≥0，o≥0)

√⊙√⊙= √( a≥0，o≥0)

Four. Simplest quadratic radical

Condition: (1) The number of square roots does not contain denominator; (2) The number of square roots does not include the factor that can be opened to the maximum.

Addition and subtraction of quadratic root

Firstly, the quadratic root is transformed into the simplest quadratic root, and then the roots with the same number are merged.

Note: Quadratic radical has double non-negative properties.