How to remove the denominator of a fractional equation

How to remove the denominator of a fractional equation

In mathematics learning, fractional equations are an important knowledge point, and how to effectively remove the denominator is a key step in solving fractional equations. This article will explain in detail the method of removing the denominator of a fractional equation, and attach the hot topics and data from the entire Internet in the past 10 days to help readers better understand this knowledge point.

1. Basic method of removing denominator from fractional equations

The core idea of denominating a fractional equation is to convert the equation into an integer equation by multiplying it by the least common multiple (LCM) of the denominator. Here are the specific steps:

1.Determine the lowest common multiple of the denominator: Find the least common multiple of all denominators, which is the basis for denominator removal.

2.Multiply both sides of the equation by the least common multiple: Convert the fractional equation into an integer equation by eliminating the denominator through the multiplication operation.

3.Solve integral equations: Solve for the value of the unknown according to the method of solving integral equations.

4.Check the plausibility of the solution: Since removing the denominator may lead to increased roots, it is necessary to check whether the solution satisfies the original equation.

2. Hot topics on the Internet in the past 10 days

The following are hot topics that have attracted much attention across the Internet in the past 10 days for readers’ reference:

3. Example analysis of removing denominator from fractional equation

In order to better understand the method of removing the denominator of a fractional equation, let us illustrate it through a specific example:

Example questions: Solve the equation (frac{2}{x} + frac{3}{x+1} = 1).

1.Determine the lowest common multiple of the denominator: The denominators are (x) and (x+1), and the least common multiple is (x(x+1)).

2.Multiply both sides of the equation by the least common multiple:

[x(x+1) cdot left( frac{2}{x} + frac{3}{x+1} right) = x(x+1) cdot 1]

After simplification we get:

[2(x+1) + 3x = x(x+1)]

3.Solve integral equations: Expand and organize equations:

[2x + 2 + 3x = x^2 + x]

[5x + 2 = x^2 + x]

Put the equation into standard form:

[x^2 - 4x - 2 = 0]

Use the root formula to solve:

[x = 2 pm sqrt{6}]

4.Check the plausibility of the solution: Verify whether (x = 2 pm sqrt{6}) makes the denominator of the original equation zero. If not, it is a valid solution.

4. Common mistakes and precautions

In the process of removing the denominator of a fractional equation, the following errors are prone to occur:

1.Ignore calculation of least common multiple: Incorrectly choosing a common multiple may result in failure to completely eliminate the denominator.

2.Forgot to check for root increase: Added roots may be introduced after removing the denominator, and the rationality of the solution must be tested.

3.Symbol error: In multiplication operations, it is easy to ignore the change of sign, leading to equation errors.

5. Summary

Removing the denominator of a fractional equation is an important step in solving a fractional equation. With the correct methods and steps, the fractional equation can be effectively converted into an integral equation to solve the unknowns. At the same time, testing the rationality of the solution is the key to avoiding root increase. I hope the explanations and examples in this article can help readers master this knowledge point.

In addition, the hot topics on the Internet in the past 10 days also reflect the current focus of society. Readers can combine mathematics learning with social hot spots to broaden their knowledge horizons.