- the formation of the concept of fractional rational equations,
- consider various ways of solving fractional rational equations,
- consider an algorithm for solving fractional rational equations, including the condition that the fraction is zero,
- to teach the solution of fractional rational equations according to an algorithm,
- checking the level of mastering the topic by conducting test work.
- development of the ability to correctly operate on the acquired knowledge, think logically,
- development of intellectual skills and mental operations - analysis, synthesis, comparison and generalization,
- development of initiative, ability to make decisions, do not stop there,
- the development of critical thinking,
- development of research skills.
- education of cognitive interest in the subject,
- fostering independence in solving educational problems,
- nurturing will and perseverance to achieve final results.
Lesson type: lesson - explanation of new material.
During the classes
1. Organizational moment.
Hello guys! Equations are written on the board. Look at them carefully. Can you solve all of these equations? Which are not and why?
Equations in which the left and right sides are fractional rational expressions are called fractional rational equations. What do you think we will learn today in the lesson? State the topic of the lesson. So, we open notebooks and write down the theme of the lesson “Solving fractional rational equations”.
2. Updating knowledge. Frontal survey, oral work with the class.
And now we will repeat the basic theoretical material that we need to study a new topic. Please answer the following questions:
- What is an equation? (Equal to Variable or Variables.)
- What is the name of the equation number 1? (Linear.) A method for solving linear equations. (Transfer everything with the unknown to the left side of the equation, all numbers to the right. Give similar terms. Find Unknown Multiplier).
- What is the name of the equation number 3? (Square.) Methods for solving quadratic equations. (Isolation of a full square, by formulas, using the Vieta theorem and its corollaries.)
- What is the proportion? (Equality of two relations.) The main property of proportion. (If the proportion is correct, then the product of its extreme members is equal to the product of the middle members.)
- What properties are used in solving equations? (1. If you transfer the term from one part to another in the equation by changing its sign, you get an equation equivalent to this one. 2. If both parts of the equation are multiplied or divided by the same non-zero number, then we get an equation equivalent to this.)
- When is the fraction equal to zero? (A fraction is zero when the numerator is zero and the denominator is not zero.)
3. Explanation of the new material.
Solve equation 2 in notebooks and on the board.
What fractional rational equation can you try to solve using the main property of proportion? (No. 5).
x 2 -4x-2x + 8 = x 2 + 3x + 2x + 6
x 2 -6x-x 2 -5x = 6-8
Solve equation 4 in notebooks and on the blackboard.
What fractional rational equation can you try to solve by multiplying both sides of the equation by the denominator? (No. 6).
Now try to solve equation No. 7 in one of the ways.
Rational Expressions and Rational Equations
We have already learned how to solve quadratic equations. Now we extend the studied methods to rational equations.
What is a rational expression? We have already come across this concept. Rational expressions Expressions are made up of numbers, variables, their degrees and signs of mathematical actions.
Accordingly, rational equations are called equations of the form: - rational expressions.
Previously, we considered only those rational equations that reduce to linear ones. Now we will consider those rational equations that are reduced to square ones.
An example of solving a rational equation
Solve the equation:.
At the very beginning, we transfer all the terms to the left, so that 0 remains on the right. We get:
Now we bring the left side of the equation to a common denominator:
A fraction is 0 if and only if its numerator is 0 and the denominator is not 0.
We get the following system:
The first equation of the system is the quadratic equation. Before solving it, divide all its coefficients by 3. We get:
The coefficients of this equation:
Further, by the formula of the roots of the quadratic equation, we find:
We get two roots:.
Now we solve the second inequality: the product of factors is not equal to 0 if and only if none of the factors is equal to 0.
Since 2 is never equal to 0, it is necessary that two conditions are satisfied:. Since none of the roots of the equation obtained above coincides with the invalid values of the variable that were obtained when solving the second inequality, they are both solutions to this equation.
Algorithm for solving a rational equation
So, let's formulate an algorithm for solving rational equations:
1. Move all the terms to the left side, so that on the right side it turns out 0.
2. Convert and simplify the left side, bring all fractions to a common denominator.
3. Equate the obtained fraction to 0, according to the following algorithm:.
4. Write down the roots that turned out in the first equation and satisfy the second inequality in response.