Complex equation with six-story fractions. ODZ. Valid Range
"Solution of fractional rational equations"
Lesson Objectives:
Tutorial:
- formation of the concept of fractional rational equations; to consider various ways of solving fractional rational equations; consider an algorithm for solving fractional rational equations, including the condition that the fraction is equal to zero; to teach the solution of fractional rational equations according to the algorithm; checking the level of assimilation of the topic by conducting test work.
Developing:
- development of the ability to correctly operate with the acquired knowledge, to think logically; development of intellectual skills and mental operations- analysis, synthesis, comparison and generalization; development of initiative, the ability to make decisions, not to stop there; development of critical thinking; development of research skills.
Nurturing:
- education of cognitive interest in the subject; education of independence in decision learning objectives; education of will and perseverance to achieve the final results.
Lesson type: lesson - explanation of new material.
During the classes
1. Organizational moment.
Hello guys! Equations are written on the blackboard, look at them carefully. Can you solve all of these equations? Which ones 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 study today in the lesson? Formulate the topic of the lesson. So, we open notebooks and write down the topic of the lesson “Solution of fractional rational equations”.
2. Actualization of knowledge. Frontal survey, oral work with the class.
And now we will repeat the main theoretical material that we need to study new topic. Please answer the following questions:
1. What is an equation? ( Equality with a variable or variables.)
2. What is Equation #1 called? ( Linear.) Method of solution linear equations. (Move everything with the unknown to the left side of the equation, all numbers to the right. Bring like terms. Find the unknown multiplier).
3. What is Equation #3 called? ( Square.) Ways to solve quadratic equations. (Selection of the full square, by formulas, using the Vieta theorem and its consequences.)
4. What is a proportion? ( Equality of two relations.) The main property of proportion. ( If the proportion is true, then the product of its extreme terms is equal to the product of the middle terms.)
5. What properties are used in solving equations? ( 1. If in the equation we transfer the term from one part to another, changing its sign, then we get an equation equivalent to the given one. 2. If both parts of the equation are multiplied or divided by the same non-zero number, then an equation will be obtained that is equivalent to the given.)
6. When is a fraction equal to zero? ( The fraction is zero when the numerator zero, and the denominator is not equal to zero.)
3. Explanation of new material.
Solve equation No. 2 in notebooks and on the board.
Answer: 10.
What fractional rational equation can you try to solve using the basic property of proportion? (No. 5).
(x-2)(x-4) = (x+2)(x+3)
x2-4x-2x+8 = x2+3x+2x+6
x2-6x-x2-5x = 6-8
Solve equation No. 4 in notebooks and on the board.
Answer: 1,5.
What fractional rational equation can you try to solve by multiplying both sides of the equation by the denominator? (No. 6).
D=1>0, x1=3, x2=4.
Answer: 3;4.
Now try to solve equation #7 in one of the ways.
(x2-2x-5)x(x-5)=x(x-5)(x+5) | |||
(x2-2x-5)x(x-5)-x(x-5)(x+5)=0 | |||
x(x-5)(x2-2x-5-(x+5))=0 | x2-2x-5-x-5=0 |
||
x(x-5)(x2-3x-10)=0 | |||
x=0 x-5=0 x2-3x-10=0 | |||
x1=0 x2=5 D=49 | |||
Answer: 0;5;-2. | Answer: 5;-2. |
Explain why this happened? Why are there three roots in one case and two in the other? What numbers are the roots of this fractional rational equation?
Until now, students have not met the concept of an extraneous root, it is really very difficult for them to understand why this happened. If no one in the class can give a clear explanation of this situation, then the teacher asks leading questions.
- How do equations No. 2 and 4 differ from equations No. 5,6,7? ( In equations No. 2 and 4 in the denominator of the number, No. 5-7 - expressions with a variable.) What is the root of the equation? ( The value of the variable at which the equation becomes a true equality.) How to find out if the number is the root of the equation? ( Make a check.)
When doing a test, some students notice that they have to divide by zero. They conclude that the numbers 0 and 5 are not the roots of this equation. The question arises: is there a way to solve fractional rational equations that eliminates this error? Yes, this method is based on the condition that the fraction is equal to zero.
x2-3x-10=0, D=49, x1=5, x2=-2.
If x=5, then x(x-5)=0, so 5 is an extraneous root.
If x=-2, then x(x-5)≠0.
Answer: -2.
Let's try to formulate an algorithm for solving fractional rational equations in this way. Children themselves formulate the algorithm.
Algorithm for solving fractional rational equations:
1. Move everything to the left side.
2. Bring fractions to a common denominator.
3. Make a system: the fraction is equal to zero when the numerator is equal to zero, and the denominator is not equal to zero.
4. Solve the equation.
5. Check the inequality to exclude extraneous roots.
6. Write down the answer.
Discussion: how to formalize the solution if the basic property of proportion is used and the multiplication of both sides of the equation by common denominator. (Supplement the solution: exclude from its roots those that turn the common denominator to zero).
4. Primary comprehension of new material.
Work in pairs. Students choose how to solve the equation on their own, depending on the type of equation. Tasks from the textbook "Algebra 8", 2007: No. 000 (b, c, i); No. 000(a, e, g). The teacher controls the performance of the task, answers the questions that have arisen, and provides assistance to poorly performing students. Self-test: Answers are written on the board.
b) 2 is an extraneous root. Answer:3.
c) 2 is an extraneous root. Answer: 1.5.
a) Answer: -12.5.
g) Answer: 1; 1.5.
5. Statement of homework.
2. Learn the algorithm for solving fractional rational equations.
3. Solve in notebooks No. 000 (a, d, e); No. 000(g, h).
4. Try to solve No. 000(a) (optional).
6. Fulfillment of the control task on the studied topic.
The work is done on sheets.
Job example:
A) Which of the equations are fractional rational?
B) A fraction is zero when the numerator is ______________________ and the denominator is _______________________.
Q) Is the number -3 the root of Equation #6?
D) Solve equation No. 7.
Task evaluation criteria:
- "5" is given if the student completed more than 90% of the task correctly. "4" - 75% -89% "3" - 50% -74% "2" is given to the student who completed less than 50% of the task. Grade 2 is not put in the journal, 3 is optional.
7. Reflection.
On the leaflets with independent work, put:
- 1 - if the lesson was interesting and understandable to you; 2 - interesting, but not clear; 3 - not interesting, but understandable; 4 - not interesting, not clear.
8. Summing up the lesson.
So, today in the lesson we got acquainted with fractional rational equations, learned how to solve these equations different ways, tested their knowledge with the help of training independent work. You will learn the results of independent work in the next lesson, at home you will have the opportunity to consolidate the knowledge gained.
What method of solving fractional rational equations, in your opinion, is easier, more accessible, more rational? Regardless of the method of solving fractional rational equations, what should not be forgotten? What is the "cunning" of fractional rational equations?
Thank you all, the lesson is over.
Equations containing a variable in the denominator can be solved in two ways:
Reducing fractions to a common denominator
Using the basic property of proportion
Regardless of the method chosen, after finding the roots of the equation, it is necessary to choose from the found values the acceptable values, i.e. those that do not turn the denominator to $0$.
1 way. Bringing fractions to a common denominator.
Example 1
$\frac(2x+3)(2x-1)=\frac(x-5)(x+3)$
Solution:
1. Move the fraction from the right side of the equation to the left
\[\frac(2x+3)(2x-1)-\frac(x-5)(x+3)=0\]
In order to do this correctly, we recall that when moving elements to another part of the equation, the sign in front of the expressions changes to the opposite. So, if on the right side there was a “+” sign before the fraction, then on the left side there will be a “-” sign in front of it. Then on the left side we get the difference of the fractions.
2. Now we note that the fractions have different denominators, which means that in order to make up the difference, it is necessary to bring the fractions to a common denominator. The common denominator will be the product of the polynomials in the denominators of the original fractions: $(2x-1)(x+3)$
In order to obtain an identical expression, the numerator and denominator of the first fraction must be multiplied by the polynomial $(x+3)$, and the second by the polynomial $(2x-1)$.
\[\frac((2x+3)(x+3))((2x-1)(x+3))-\frac((x-5)(2x-1))((x+3)( 2x-1))=0\]
Let's perform the transformation in the numerator of the first fraction - we will multiply the polynomials. Recall that for this it is necessary to multiply the first term of the first polynomial, multiply by each term of the second polynomial, then multiply the second term of the first polynomial by each term of the second polynomial and add the results
\[\left(2x+3\right)\left(x+3\right)=2x\cdot x+2x\cdot 3+3\cdot x+3\cdot 3=(2x)^2+6x+3x +9\]
We present similar terms in the resulting expression
\[\left(2x+3\right)\left(x+3\right)=2x\cdot x+2x\cdot 3+3\cdot x+3\cdot 3=(2x)^2+6x+3x +9=\] \[(=2x)^2+9x+9\]
Perform a similar transformation in the numerator of the second fraction - we will multiply the polynomials
$\left(x-5\right)\left(2x-1\right)=x\cdot 2x-x\cdot 1-5\cdot 2x+5\cdot 1=(2x)^2-x-10x+ 5=(2x)^2-11x+5$
Then the equation will take the form:
\[\frac((2x)^2+9x+9)((2x-1)(x+3))-\frac((2x)^2-11x+5)((x+3)(2x- 1))=0\]
Now fractions with same denominator, so you can do the subtraction. Recall that when subtracting fractions with the same denominator from the numerator of the first fraction, it is necessary to subtract the numerator of the second fraction, leaving the denominator the same
\[\frac((2x)^2+9x+9-((2x)^2-11x+5))((2x-1)(x+3))=0\]
Let's transform the expression in the numerator. In order to open the brackets preceded by the “-” sign, all signs in front of the terms in brackets must be reversed
\[(2x)^2+9x+9-\left((2x)^2-11x+5\right)=(2x)^2+9x+9-(2x)^2+11x-5\]
We present like terms
$(2x)^2+9x+9-\left((2x)^2-11x+5\right)=(2x)^2+9x+9-(2x)^2+11x-5=20x+4 $
Then the fraction will take the form
\[\frac((\rm 20x+4))((2x-1)(x+3))=0\]
3. A fraction is equal to $0$ if its numerator is 0. Therefore, we equate the numerator of the fraction to $0$.
\[(\rm 20x+4=0)\]
Let's solve the linear equation:
4. Let's sample the roots. This means that it is necessary to check whether the denominators of the original fractions turn into $0$ when the roots are found.
We set the condition that the denominators are not equal to $0$
x$\ne 0.5$ x$\ne -3$
This means that all values of the variables are allowed, except for $-3$ and $0.5$.
The root we found is a valid value, so it can be safely considered the root of the equation. If the found root were not a valid value, then such a root would be extraneous and, of course, would not be included in the answer.
Answer:$-0,2.$
Now we can write an algorithm for solving an equation that contains a variable in the denominator
An algorithm for solving an equation that contains a variable in the denominator
Move all elements from the right side of the equation to the left side. To obtain an identical equation, it is necessary to change all the signs in front of the expressions on the right side to the opposite
If on the left side we get an expression with different denominators, then we bring them to the general one, using the main property of the fraction. Perform transformations using identical transformations and get the final fraction equal to $0$.
Equate the numerator to $0$ and find the roots of the resulting equation.
Let's sample the roots, i.e. find valid variable values that do not turn the denominator to $0$.
2 way. Using the basic property of proportion
The main property of a proportion is that the product of the extreme terms of the proportion is equal to the product of the middle terms.
Example 2
We use given property to solve this task
\[\frac(2x+3)(2x-1)=\frac(x-5)(x+3)\]
1. Let's find and equate the product of the extreme and middle members of the proportion.
$\left(2x+3\right)\cdot(\ x+3)=\left(x-5\right)\cdot(2x-1)$
\[(2x)^2+3x+6x+9=(2x)^2-10x-x+5\]
Solving the resulting equation, we find the roots of the original
2. Let's find admissible values of a variable.
From the previous solution (1st way) we have already found that any values are allowed except $-3$ and $0.5$.
Then, having established that the found root is a valid value, we found out that $-0.2$ will be the root.
Fraction calculator designed for quick calculation of operations with fractions, it will help you easily add, multiply, divide or subtract fractions.
Modern schoolchildren begin to study fractions already in the 5th grade, and every year the exercises with them become more complicated. Mathematical terms and quantities that we learn in school are rarely useful to us in adulthood. However, fractions, unlike logarithms and degrees, are quite common in everyday life (measuring distance, weighing goods, etc.). Our calculator is designed for quick operations with fractions.
First, let's define what fractions are and what they are. Fractions are the ratio of one number to another; this is a number consisting of a whole number of fractions of a unit.
Fraction types:
- Ordinary
- Decimals
- mixed
Example ordinary fractions:
The top value is the numerator, the bottom is the denominator. The dash shows us that the top number is divisible by the bottom number. Instead of a similar writing format, when the dash is horizontal, you can write differently. You can put a slanted line, for example:
1/2, 3/7, 19/5, 32/8, 10/100, 4/1
Decimals are the most popular type of fractions. They consist of an integer part and a fractional part, separated by a comma.
Decimal example:
0.2 or 6.71 or 0.125
It consists of an integer and a fractional part. To find out the value of this fraction, you need to add the whole number and the fraction.
Example of mixed fractions:
The fraction calculator on our website is able to quickly perform any mathematical operations with fractions:
- Addition
- Subtraction
- Multiplication
- Division
To carry out the calculation, you need to enter the numbers in the fields and select the action. For fractions, you need to fill in the numerator and denominator, an integer may not be written (if the fraction is ordinary). Don't forget to click on the "equal" button.
It is convenient that the calculator immediately provides a process for solving an example with fractions, and not just a ready-made answer. It is thanks to the detailed solution that you can use this material in solving school problems and for better mastering the material covered.
You need to calculate the example:
After entering the indicators in the form fields, we get:
To make an independent calculation, enter the data in the form.
Fraction calculator
Enter two fractions:+ - * : | |||||||
related sections.
Lesson Objectives:
Tutorial:
- formation of the concept of fractional rational equations;
- to consider various ways of solving fractional rational equations;
- consider an algorithm for solving fractional rational equations, including the condition that the fraction is equal to zero;
- to teach the solution of fractional rational equations according to the algorithm;
- checking the level of assimilation of the topic by conducting test work.
Developing:
- development of the ability to correctly operate with the acquired knowledge, to think logically;
- development of intellectual skills and mental operations - analysis, synthesis, comparison and generalization;
- development of initiative, the ability to make decisions, not to stop there;
- development of critical thinking;
- development of research skills.
Nurturing:
- education of cognitive interest in the subject;
- education of independence in solving educational problems;
- education of will and perseverance to achieve the final results.
Lesson type: lesson - explanation of new material.
During the classes
1. Organizational moment.
Hello guys! Equations are written on the blackboard, look at them carefully. Can you solve all of these equations? Which ones 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 study today in the lesson? Formulate the topic of the lesson. So, we open notebooks and write down the topic of the lesson “Solution of fractional rational equations”.
2. Actualization of knowledge. Frontal survey, oral work with the class.
And now we will repeat the main theoretical material that we need to study a new topic. Please answer the following questions:
- What is an equation? ( Equality with a variable or variables.)
- What is equation #1 called? ( Linear.) Method for solving linear equations. ( Move everything with the unknown to the left side of the equation, all numbers to the right. Bring like terms. Find the unknown multiplier).
- What is Equation 3 called? ( Square.) Methods for solving quadratic equations. ( Selection of the full square, by formulas, using the Vieta theorem and its consequences.)
- What is a proportion? ( Equality of two relations.) The main property of proportion. ( If the proportion is true, then the product of its extreme terms is equal to the product of the middle terms.)
- What properties are used to solve equations? ( 1. If in the equation we transfer the term from one part to another, changing its sign, then we get an equation equivalent to the given one. 2. If both parts of the equation are multiplied or divided by the same non-zero number, then an equation will be obtained that is equivalent to the given.)
- When is a fraction equal to zero? ( A fraction is zero when the numerator is zero and the denominator is non-zero.)
3. Explanation of new material.
Solve equation No. 2 in notebooks and on the board.
Answer: 10.
What fractional rational equation can you try to solve using the basic property of proportion? (No. 5).
(x-2)(x-4) = (x+2)(x+3)
x 2 -4x-2x + 8 \u003d x 2 + 3x + 2x + 6
x 2 -6x-x 2 -5x \u003d 6-8
Solve equation No. 4 in notebooks and on the board.
Answer: 1,5.
What fractional rational equation can you try to solve by multiplying both sides of the equation by the denominator? (No. 6).
x 2 -7x+12 = 0
D=1>0, x 1 =3, x 2 =4.
Answer: 3;4.
Now try to solve equation #7 in one of the ways.
(x 2 -2x-5)x(x-5)=x(x-5)(x+5) |
|||
(x 2 -2x-5)x(x-5)-x(x-5)(x+5)=0 |
x 2 -2x-5=x+5 |
||
x(x-5)(x 2 -2x-5-(x+5))=0 |
x 2 -2x-5-x-5=0 |
||
x(x-5)(x 2 -3x-10)=0 |
|||
x=0 x-5=0 x 2 -3x-10=0 |
|||
x 1 \u003d 0 x 2 \u003d 5 D \u003d 49 |
|||
x 3 \u003d 5 x 4 \u003d -2 |
x 3 \u003d 5 x 4 \u003d -2 |
||
Answer: 0;5;-2. |
Answer: 5;-2. |
Explain why this happened? Why are there three roots in one case and two in the other? What numbers are the roots of this fractional rational equation?
Until now, students have not met the concept of an extraneous root, it is really very difficult for them to understand why this happened. If no one in the class can give a clear explanation of this situation, then the teacher asks leading questions.
- How do equations No. 2 and 4 differ from equations No. 5,6,7? ( In equations No. 2 and 4 in the denominator of the number, No. 5-7 - expressions with a variable.)
- What is the root of the equation? ( The value of the variable at which the equation becomes a true equality.)
- How to find out if a number is the root of an equation? ( Make a check.)
When doing a test, some students notice that they have to divide by zero. They conclude that the numbers 0 and 5 are not the roots of this equation. The question arises: is there a way to solve fractional rational equations that eliminates this error? Yes, this method is based on the condition that the fraction is equal to zero.
x 2 -3x-10=0, D=49, x 1 =5, x 2 = -2.
If x=5, then x(x-5)=0, so 5 is an extraneous root.
If x=-2, then x(x-5)≠0.
Answer: -2.
Let's try to formulate an algorithm for solving fractional rational equations in this way. Children themselves formulate the algorithm.
Algorithm for solving fractional rational equations:
- Move everything to the left.
- Bring fractions to a common denominator.
- Make up a system: a fraction is zero when the numerator is zero and the denominator is not zero.
- Solve the equation.
- Check inequality to exclude extraneous roots.
- Write down the answer.
Discussion: how to formalize the solution if the basic property of proportion is used and the multiplication of both sides of the equation by a common denominator. (Supplement the solution: exclude from its roots those that turn the common denominator to zero).
4. Primary comprehension of new material.
Work in pairs. Students choose how to solve the equation on their own, depending on the type of equation. Tasks from the textbook "Algebra 8", Yu.N. Makarychev, 2007: No. 600 (b, c, i); No. 601(a, e, g). The teacher controls the performance of the task, answers the questions that have arisen, and provides assistance to poorly performing students. Self-test: Answers are written on the board.
b) 2 is an extraneous root. Answer:3.
c) 2 is an extraneous root. Answer: 1.5.
a) Answer: -12.5.
g) Answer: 1; 1.5.
5. Statement of homework.
- Read item 25 from the textbook, analyze examples 1-3.
- Learn the algorithm for solving fractional rational equations.
- Solve in notebooks No. 600 (a, d, e); No. 601 (g, h).
- Try to solve #696(a) (optional).
6. Fulfillment of the control task on the studied topic.
The work is done on sheets.
Job example:
A) Which of the equations are fractional rational?
B) A fraction is zero when the numerator is ______________________ and the denominator is _______________________.
Q) Is the number -3 the root of Equation #6?
D) Solve equation No. 7.
Task evaluation criteria:
- "5" is given if the student completed more than 90% of the task correctly.
- "4" - 75% -89%
- "3" - 50% -74%
- "2" is given to a student who completed less than 50% of the task.
- Grade 2 is not put in the journal, 3 is optional.
7. Reflection.
On the leaflets with independent work, put:
- 1 - if the lesson was interesting and understandable to you;
- 2 - interesting, but not clear;
- 3 - not interesting, but understandable;
- 4 - not interesting, not clear.
8. Summing up the lesson.
So, today in the lesson we got acquainted with fractional rational equations, learned how to solve these equations in various ways, tested our knowledge with the help of educational independent work. You will learn the results of independent work in the next lesson, at home you will have the opportunity to consolidate the knowledge gained.
What method of solving fractional rational equations, in your opinion, is easier, more accessible, more rational? Regardless of the method of solving fractional rational equations, what should not be forgotten? What is the "cunning" of fractional rational equations?
Thank you all, the lesson is over.