What is the concept of homogeneous quadratic radicals

The number of squares is the same after being formed into the simplest quadratic radical. Such a quadratic radical is called a homogeneous quadratic radical. One quadratic radical cannot be called a homogeneous quadratic radical, but at least two quadratic radicals can be called homogeneous quadratic radicals.

To determine whether several radicals are the same kind of quadratic radicals, you must first reduce the number in the root number, and then turn the non-simplest quadratic radical into the simplest quadratic radical, and then judge.

The same quadratic radical defines "the three steps of teaching."

1) Introduce the same kind of quadratic root definition into the example, and give positive and negative examples to understand repeatedly;

2) Define the application, fully understand "after simplification, the quadratic radical with the same number of squares", and give several sets of examples that are not the simplest quadratic radicals to understand;

3) Broadening of the definition, from the definition of the same kind of quadratic radicals, the definition of the general similar radical is found (the text of the new textbook is not required).

The same quadratic radical is a special equation whose analytic form contains two quadratic radicals that are identical to it, i.e., the radical structure is identical. It is in the form of:

ax^2+bx+c=0

where a, b, and c are real numbers, and x is unknown.

Quadratic radical simplification general steps:

Turn fractions or decimals into false fractions.

Factoring the square number into prime factors or factoring.

Move the factors or factors that can be opened as far as possible in the root sign outside the root number.

Remove the denominator from the root number, or remove the root number from the denominator.

Approximation. There are physicochemical factors.

If two algebraic formulas containing quadratic radicals are multiplied, and if their product does not contain quadratic radicals, then the two algebraic equations are called mutually rational factors.

A homogeneous quadratic radical is a quadratic radical that has the same radical part. A quadratic radical is an expression that contains a variable under the root number, such as x, (2x + 1), and so on. When the lower part of the root sign of two quadratic radicals is the same, they belong to the same type of quadratic radicals.

This concept derives from the classification and comparison of radicals in mathematics. In the problems of simplification, operation, and evaluation, classifying the same quadratic radical can be convenient for merging, separating, and computation.

Application of knowledge points:

When you encounter a situation where you need to merge or compare quadratic radicals, you can tell if they are homogeneous quadratic radicals. The same quadratic radical can be used for addition, subtraction, multiplication and division, etc., as well as simplification or size comparison.

By identifying the same kind of quadratic radicals, the radical expressions can be simplified, the operation steps can be simplified, and the problem can be made more concise and easy to decode.

Explanation of knowledge points and example questions:

Example 1: Determine if the following quadratic radicals are of the same kind: 3 and 2 3

Analysis: Both of these quadratic radicals are 3 in the lower part of the root number, so they belong to the same kind of quadratic radicals.

Example 2: Simplify the following quadratic radical: 3 5 + 5

Analysis: The lower part of the root sign of these two quadratic radicals is the same, both are 5, so they are the same kind of quadratic radicals. Merging is possible: 3 5 + 5 = 3 + 1) 5 = 4 5

Example 3: Compare the size of the following quadratic radicals: 7 and 2 6

Analysis: The lower part of the root number of these two quadratic radicals is different, which is 7 and 6, respectively, so they are not the same quadratic radicals. In this case, you cannot directly compare the size, and further operations are required.

The above is the definition, application, and example solution of the same quadratic radical. By identifying homogeneous quadratic radicals, it is easier to merge, calculate, and compare radicals.