Homogeneous quadratic radicals, what are homogeneous quadratic radicals

After several quadratic radicals are reduced to the simplest quadratic radicals, if the number of squares is the same, these quadratic radicals are called the same quadratic radicals. 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.

Similarities and differences between homogeneous quadratic radicals and homogeneous terms.

The same kind of quadratic radical and the same kind of term are very similar in terms of expression and algorithm, so we list the differences and connections between the two, and pay attention to the analysis and comparison when learning.

Similarities. 1.Both are a relationship between two algebraic formulas.

A similar term is a relationship between two individual terms, and the letter and the exponent of the same letter are the same; The same quadratic radical is the relationship between two quadratic radicals, which refers to the quadratic radical with the same number of squares after being converted into the simplest quadratic radical.

2.Both can be combined, and the rules of merger are the same. If the root part of the simplest quadratic radical is regarded as the exponential part of the same term, and the factor outside the root number is regarded as the coefficient part of the same term, then the merger rule of the same quadratic radical is the same as that of the same quadratic term, that is, "the same quadratic radical (or similar term) is added or subtracted, and the root (letter) is unchanged, and the coefficients are added and subtracted".

Differences. 1.The judging criteria are different.

To determine whether two simplest quadratic radicals are the same kind of quadratic radicals, the basis is "whether the number of squares to be opened is the same", which has nothing to do with the factors outside the root number. The judgment of similar terms is based on "whether the letter factor and its exponent correspond to the same", which has nothing to do with coefficients.

2.The form of merger is different.

What is homogeneous quadratic radical.

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.

Definition: After being transformed into the simplest state to accompany the quadratic radical, the number of Kai squares is the same. Such a quadratic radical is called a homogeneous quadratic radical.

Properties: One quadratic radical cannot be called the same kind of quadratic radical, at least two quadratic radicals can be called the same kind of quadratic radical repentance. [To determine whether several radicals are the same kind of quadratic radicals, you must first simplify, turn the non-simplest quadratic radicals into the simplest quadratic radicals, and then judge. 】

Examples. Which of the following equations are homogeneous quadratic radicals?

Analysis: Comment: The key to judging whether several quadratic radicals are the same kind of quadratic radicals is to simplify them first, and the quadratic radicals with exactly the same number of squares after simplification are the same quadratic radicals

Hopefully, thank you.

After the simplest quadratic radical is formed, the parts under the root number are identical.

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 quadratic radicals of the same kind, it is easier to merge, calculate, and compare radicals.