Jumat, 26 November 2010

QUADRATIC EQUATIONS.....FRACTION

Example 1: Solve the following equation.

1 - 1 / (x - 2) = 4
Solution to example 1

  • This equation has a rational expression. Multiply all the terms of the equation by the denominator in the rational expression

    (x - 2)(1 - 1 / (x - 2)) = (x - 2)4

  • Simplify.
    (x - 2) - 1 = 4 (x - 2)

  • Mutliply factors and group like terms.
    x - 3 = 4x - 8

  • Add -4x to both sides and simplify

    -3x - 3 = - 8

  • Add + 3 to both sides and simplify

    -3x = - 5

  • Solve for x

    x = 5 /3
conclusion:The solution to the above equation is x = 5 / 3. Example 2: Find all real solutions to the equation.

1 - 1 / (x - 2) = -4 / (x 2 - 4)
Solution to example 2

  • This is an equation with two rational expressions. Multiply all the terms on the left side of the equation and all the terms in the right side of the equation by the lowest common denominator (x - 2)(x + 2).

    (x - 2)(x + 2)(1 - 1 / (x - 2)) = (x - 2)(x + 2)( -4 / (x2 - 4) )

  • Cancel common factors.
    (x - 2) (x + 2) - (x + 2) = - 4

  • multiply factors and group like terms.
    x 2 - x - 6 = -4

  • add 4 to both sides.
    x 2 - x - 2 = 0

  • factor left side of equation.
    (x + 1) (x - 2) = 0

  • 2 values make the LS zero.

    x1 = -1
    x2 = 2

  • check:
    x = -1
    Left Side of equation = 1 -1 / (-3)
    = 4/3

    Right Side of Equation = -4 / (1-4)

    = 4/3

    x = 2

    this value of x is not a solution because it makes each of the denominators in the given equation equal to zero.
conclusion:The solution to the above equation is x = -1.

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EXAMPLE PROBLEMS QUADRATICS EQUATION

Example - Problem 1: A right triangle has a perimeter of 24 cm and a hypotenuse of 10 cm. Find the sides x and y, x > y, that make the right angle of the triangle.
Solution to Problem 1:

  • We start by drawing a triangle with the given information

    Triangle to solve
  • The perimeter of the triangle is 24, hence
    x + y + 10 = 24

  • It is a right triangle, use Pythagoras theorem to obtain.
    x2 + y2 = 102

  • Solve the equation x + y + 10 = 24 for y.
    y = 14 - x

  • Substitute y in the equation x2 + y2 = 102 by the expression obtained above.
    x2 + (14 - x)2 = 102

  • Expand the square, group like terms and write the above equation with the right side equal to zero.
    2x2 -28x + 96 = 0

  • Multiply all terms in the above equation by 1/2.
    x2 -14x + 48 = 0

  • Find the discriminant of the above quadratic equation.
    Discriminant D = b2 - 4*a*c = 196 - 192 = 4

  • Use the quadratic formulas to solve the quadratic equation; two solutions
    x1 = [ -b + sqrt(D) ] / 2*a = [ 14 + 2 ] / 2 = 8

    x2 = [ -b - sqrt(D) ] / 2*a = [ 14 - 2 ] / 2 = 6

  • use the equation y = 14 - x to find the corresponding value of y.
    y1 = 14 - 8 = 6
    y2 = 14 - 6 = 8

  • Taking into account the condition x > y, the sides that make the right angle of the triangle are: x = 8 cm and y = 6 cm.

    Check answer:
    Hypotenuse h = sqrt (x2 + y2)
    = sqrt (82 cm2 + 62 cm2)
    = sqrt(64 cm2 + 36 cm2)
    = 10 cm, it agrees with the given value.

    Perimeter = y + x + hypotenuse
    = 8 cm + 6 cm + 10 cm
    = 24 cm, it agrees with the given value.
Matched Problem 1: A rectangle has a perimeter of 60 m and an area of 200 m2. Find the length x and width y, x > y, of the rectangle.
Example - Problem 2: The sum of the squares of two consecutive real numbers is 61. Find the numbers.
Solution to Problem 2:


  • Let x and x+1 be the two consecutive numbers. The sum of the square of x and x + 1 is equal to 61.
    x2 + (x + 1)2 = 61


  • Expand (x + 1)2, group like terms and write the above equation with the right side equal to zero.
    2x2 + 2x - 60 = 0


  • Multiply all terms in the above equation by 1/2.
    x2 + x - 30 = 0


  • Find the discriminant of the above quadratic equation.
    Discriminant D = b2 - 4*a*c = 1 + 120 = 121


  • Use the quadratic formulas to solve the quadratic equation; two solutions
    x1 = [ -b + sqrt(D) ] / 2*a = [ -1 + 11 ] / 2 = 5
    x2 = [ -b - sqrt(D) ] / 2*a = [ -1 - 11 ] / 2 = -6


  • First solution to the problem
    first number: x1 = 5

    second number: x1 + 1 = 6


  • Second solution to the problem
    first number: x2 = -6

    second number: x2 + 1 = -5

    Check answer:
    first solution sum of squares: 52 + 62
    = 25 + 36 = 61
    second solution sum of squares: (-6)2 + (-5)2
    = 36 + 25 = 61
    The two solutions to the problem agree with the given information in the problem.

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    SYLLABUS GRADE X-2


    SYLLABUS
    School name                                   :  First Senior High School 1 Salatiga
    Subject                                            :  Mathematics
    Grade/ Semester                             :  X/ 2
         Standard  Of Competence            : Using the logic of mathematics in problem solving that related to the compound statements (proposition) and the quantifier  statements.
    Basic Competency

    Indicator
    Materials
    Learning activities
    Time Allotment

    Sources

    Assessment

    Methods
    Instrument

    1
    2
    3
    4
    5
    6
    7
    8

    *      4.1
    To determine the truth value of the compound statement and the quainter statement.
    Þ    To determine the truth value of the quainter statement.
    Þ    To determine the negation of the quainter statement.
    Þ    To determine the truth value of the compound statement.
    Þ    To determine the negation of the compound statement..
    Þ     Logic of mathematics.
    Þ    The statement and its truth value.
    Þ    The quainter statement
    Þ    The negation of the statement
    Þ    The com-pound statement
    (The truth value and its negation).
    Þ    Differentiating the statement and not the statement.
    Þ    Determining the truth value of statement.
    Þ    Determining the negation of statement.
    Þ    Identifying the  compound statement characteristic in form of conjunction, disjunction, implication and biimplication.
    Þ    Formulating the truth value  of the compound statement in the form of disjunction, implication and biimplication.
    Þ    Determining the truth value of the compound statement in the form of disjunction, implication and biimplication.
    Þ    Formulating the negation of  the compound statement in the form of disjunction, implication and biimplication by using the truth value table.
    ·         2 x 45 minutes



    ·         4 x 45 minutes
    Ø  Sources:
    ·         Text book (Math- Material Addition by Wahyu Tri Astuti.
    ·         Reference book *(Mathematics Revision Guide (Martin Law). * Additional Mathematics ( Ang Tok Woon). * Mathematics IGCSE (Ric Pimentel).
    Ø  Teaching aids:
    ·         LCD
    ·         Note book
    Ø  Individual Task
    Ø  Group Task
    Ø  Examination
    Ø  Quiz
    Ø  Multiple Choice test
    Ø  Matching test
    Ø  Essay test

    Basic Competency

    Indicator
    Materials
    Learning activities
    Time Allotment

    Sources

    Assessment

    Methods
    Instrument
    1
    2
    3
    4
    5
    6
    7
    8


    ·    Conjunctions
    ·    Disjunctions.
    ·    Implications.
    ·    Biimplications.   
    Þ    Determining the negation of the compound statement in the form of disjunction, implication and biimplication by using the truth table.
    Þ    Identifying the daily statement in daily that has connection with the compound statement.
    Þ    Identifying the connection between implication and converse, inverse and its contraposition.
    Þ    Determining the converse, inverse and contraposition of the statement in the form of implication.
    ·         2 x 45 minutes



    ·         4 x 45 minutes
    Ø  Sources:
    ·         Text book (Math- Material Addition by Wahyu Tri Astuti.
    ·         Reference book *(Mathematics Revision Guide (Martin Law). * Additional Mathematics ( Ang Tok Woon). * Mathematics IGCSE (Ric Pimentel).
    Ø  Teaching aids:
    ·         LCD
    ·         Note book
    Ø  Individual Task
    Ø  Group Task
    Ø  Examination
    Ø  Quiz
    Ø  Multiple Choice test
    Ø  Matching test
    Ø  Essay test

























    Basic Competency

    Indicator
    Materials
    Learning activities
    Time Allotment

    Sources

    Assessment

    Methods
    Instrument
    1
    2
    3
    4
    5
    6
    7
    8
    *      4.2
    To formulate the statement that equivalent with the compound statement or the quainter statement.  
    Þ    To check the equivalent between two the compound statement or the quainter statement.
    Þ    To prove the equivalent between two the compound statement or the quainter statement.
    Þ    To make the statement that equivalent with the compound statement or the quainter statement.   
    Þ    The equivalent of two compound statement.
    Þ    The tautology and contradiction.
    Þ    Identifying the compound statement that equivalent.
    Þ    Checking the equivalent between two the compound statement or the quainter statement.
    Þ    Proofing the equivalent between two the compound statement or the quainter statement.
    Þ    Identifying the characteristic of the tautology statement and the contradiction based on the truth value table.
    Þ    Checking the compound statement whether it is the  tautology or contradiction or neither.
    ·         4 x 45 minutes



    ·          
    Ø  Sources:
    ·         Text book (Math- Material Addition by Wahyu Tri Astuti.
    ·         Reference book *(Mathematics Revision Guide (Martin Law). * Additional Mathematics ( Ang Tok Woon). * Mathematics IGCSE (Ric Pimentel).
    Ø  Teaching aids:
    ·         LCD
    ·         Note book
    Ø  Individual Task
    Ø  Group Task
    Ø  Examination
    Ø  Quiz
    Ø  Multiple Choice test
    Ø  Matching test
    Ø  Essay test












    Basic Competency

    Indicator
    Materials
    Learning activities
    Time Allotment

    Sources

    Assessment

    Methods
    Instrument
    1
    2
    3
    4
    5
    6
    7
    8
    *      4.3
    Using the  logical principle that connected  to the compound statement and the quainter statement in conclusion making and problem solving. 
    Þ    To check the validity of conclusion making by using logical principle.
    Þ    To determine the conclusion from some available premise.
    ·    Drawing
    ·    conclusion.
    ·    Modus Tolens.
    ·    Modus ponens
    ·    Syllogism.


       
    Þ    Identifying the way of drawing  conclusion from some available example that is given.
    Þ    Formulating the method of drawing conclusion  based on implication. (modus ponens, modus tolens and syllogism).
    Þ    Checking the validity of drawing conclusion.
    Þ    Arranging the valid conclusion that is based on the available premise that is given.
    ·         2 x 45 minutes



    ·         4 x 45 minutes
    Ø  Sources:
    ·         Text book (Math- Material Addition by Wahyu Tri Astuti.
    ·         Reference book *(Mathematics Revision Guide (Martin Law). * Additional Mathematics ( Ang Tok Woon). * Mathematics IGCSE (Ric Pimentel).
    Ø  Teaching aids:
    ·         LCD
    ·         Note book
    Ø  Individual Task
    Ø  Group Task
    Ø  Examination
    Ø  Quiz
    Ø  Multiple Choice test
    Ø  Matching test
    Ø  Essay test











    SYLLABUS
    School name                              :  First Senior High School 1 Salatiga
    Subject                                      :  Mathematics
    Grade/ Semester                       :  X/ 2
    Standard  Of Competence       : Using the ratios, functions, equations and trigonometric identities in problem solving.
    Basic Competency

    Indicator
    Materials
    Learning activities
    Time Allotment

    Sources

    Assessment

    Methods
    Instrument

    1
    2
    3
    4
    5
    6
    7
    8

    *      5.1
    To manipulate e algebra in calculation technique that related to  the trigonometric ratios, functions, equations, and identities.
    Þ     To find the trigonometric ratio values of right triangle.
    Þ     To find the Trigonometric ratio  values from special angles
    Þ     To find the Trigonometric ratio values of angles in all quadrant
    Trigonometric
    Þ     The ratio of Trigonometric in right triangle.
    Þ     The ratios of Trigonometric from special angle.
    Þ     The ratios of Trigonometric from angles of all quadrants.
    Þ     Calculating the ratios of sides in a right triangle that its angle is constant but its length of side is different ratios values in special angles.
    Þ     Identifying the meaning of trigonometric ratio at the right triangle.
    Þ     Determining the value of trigonometric ratio of an angle to right triangle.
    Þ     Searching the value of trigonometric ratios from special angle.
    Þ     Using the trigonometric ratios value of special angle in problem solving.






    ·         10 x 45 minutes



    ·          
    Ø  Sources:
    ·         Text book (Math- Material Addition by Wahyu Tri Astuti.
    ·         Reference book *(Mathematics Revision Guide (Martin Law). * Additional Mathematics
    ·         ( Ang Tok Woon). * Mathematics IGCSE (Ric Pimentel).
    Ø  Teaching aids:
    ·         LCD
    ·         Note book
    Ø  Individual Task
    Ø  Group Task
    Ø  Examination
    Ø  Quiz
    Ø  Multiple Choice test
    Ø  Matching test
    Ø  Essay test

    Basic Competency

    Indicator
    Materials
    Learning activities
    Time Allotment

    Sources

    Assessment

    Methods
    Instrument
    1
    2
    3
    4
    5
    6
    7
    8


      
    Þ    Derivatifing the trigonometric ratios formula of an angle to cartesian plane.
    Þ    Doing the calculation of trigonometric ratio at cartesian plane.
    Þ    Searching the relation between trigonometric ratio of angles in all quadrants.
    Þ    Determining the value of trigonometric ratio of angles in various quadrants.
    ·          
    Ø  Sources:
    ·         Text book (Math- Material Addition by Wahyu Tri Astuti.
    ·         Reference book *(Mathematics Revision Guide (Martin Law). * Additional Mathematics ( Ang Tok Woon). * Mathematics IGCSE (Ric Pimentel).
    Ø  Teaching aids:
    ·         LCD
    ·         Note book
    Ø  Individual Task
    Ø  Group Task
    Ø  Examination
    Ø  Quiz
    Ø  Multiple Choice test
    Ø  Matching test
    Ø  Essay test

























    Basic Competency

    Indicator
    Materials
    Learning activities
    Time Allotment

    Sources

    Assessment

    Methods
    Instrument
    1
    2
    3
    4
    5
    6
    7
    8
    *      5.2
    To design mathematics mode from problem that related to the  functions ratios, equations and trigonometric identities.
    Þ    To plot the graph of the simple trigonometric functions in all quadrants.
    Þ    To solve the simple trigonometric equations
    Þ    To prove the simple trigonometric identities.
    Þ    To solve the calculation problem of use sine and cosine formulas.
    Þ    To determine the area of triangle that its component has been known.
    Þ    Trigonometric functions and its graphs.
    Þ    Simple trigonometric equations.
    Þ    Trigonometric identities.
    Þ    Sine formula and cosine formula.
    Þ    Area of triangle formula.
    Þ    Determining the value of trigonometric functions.
    Þ    Plotting the simple trigonometric function graph.
    Þ    Determining the solution of simple trigonometric equations.
    Þ    Formulating the relation between trigonometric ratios of an angle.
    Þ    Proofing the simple trigonometric identities by using formula and the relation between trigonometric ratios.
    Þ    Identifying the problem of side calculation or angle calculation in triangle.
    Þ    Formulating the sine formula and cosine formula.
    Þ    Using the sine formula and cosine formula for problem solving in calculation of side and angle in triangle.
    Þ    Identifying the problem in calculation of triangle area.
    Þ    Finding the formula of triangle area.
    Þ    Using the formula of triangle area for problem solving.






    ·         20 x 45 minutes



    ·          
    Ø  Sources:
    ·         Text book (Math- Material Addition by Wahyu Tri Astuti.
    ·         Reference book *(Mathematics Revision Guide (Martin Law). * Additional Mathematics ( Ang Tok Woon). * Mathematics IGCSE (Ric Pimentel).
    Ø  Teaching aids:
    ·         LCD
    ·         Note book
    Ø  Individual Task
    Ø  Group Task
    Ø  Examination
    Ø  Quiz
    Ø  Multiple Choice test
    Ø  Matching test
    Ø  Essay test
    Basic Competency

    Indicator
    Materials
    Learning activities
    Time Allotment

    Sources

    Assessment

    Methods
    Instrument
    1
    2
    3
    4
    5
    6
    7
    8
    *      5.3
    To solve the mathematical model of problem that related to  the functions ratios, equations, trigonometric identities and their interpretation.
    Þ    To identify the problem that related to the  ratios, function, equations and trigonometric identities.
    Þ    To make the mathematics model that related to the  trigonometric ratios, function, equations and trigonometric identities.
    Þ    To find the solve of mathematics model that related to the   trigonometric ratios, function, equations and trigonometric identities.
    Þ       To use trigonometric ratios.  
    Þ    Identifying the problem that related to  the trigonometric  ratios, functions, equations and trigonometric identities.
    Þ    Making the mathematical model of problem that related to the  trigonometric ratios, functions, equations and trigonometric identities.
    Þ    Solving the mathematics model from problem that related to the  trigonometric ratios, functions, equations and trigonometric identities.
    Þ    Estimating/ interpreting the result problem solve that related to the trigonometric  ratios, functions, equations and trigonometric identities.
    ·         4 x 45 minutes
    ·          
    Ø  Sources:
    ·         Text book (Math- Material Addition by Wahyu Tri Astuti.
    ·         Reference book *(Mathematics Revision Guide (Martin Law). * Additional Mathematics ( Ang Tok Woon). * Mathematics IGCSE (Ric Pimentel).
    Ø  Teaching aids:
    ·         LCD
    ·         Note book
    Ø  Individual Task
    Ø  Group Task
    Ø  Examination
    Ø  Quiz
    Ø  Multiple Choice test
    Ø  Matching test
    Ø  Essay test


















    SYLLABUS
    School name                              :  First Senior High School 1 Salatiga
    Subject                                      :  Mathematics
    Grade/ Semester                       :  X/ 2
    Standard  Of Competence       : Determining the positions, distance, and angle size that involve the point, line and plane in the three dimensional space.

    Basic Competency

    Indicator
    Materials
    Learning activities
    Time Allotment

    Sources

    Assessment

    Methods
    Instrument

    1
    2
    3
    4
    5
    6
    7
    8

    6.1 To find the point position, line and plane in the three dimensional space.
    Þ    To find the point position and line in three dimensional space.
    Þ    To find the point position and plane in three dimensional space.
    Þ    To find the position between two line in three dimensional space.
    Þ    To find the line position an plane in three dimensional space.
    Þ    To find the position between two planes in three dimensional space.
    Three Dimensional Space.
    Þ    Acquaintance three dimensional space.
    Þ    Point position three dimensional space.
    Þ    Point position, line, and plane three dimensional space.
    Þ    Identifying the objects of three dimensional space.
    Þ    Identifying the elements of three dimensional space.
    Þ    Searching the position between elements of three dimensional space.
    Þ    Describing the position between elements of three dimensional space.
    ·         4 x 45 minutes



    ·          
    Ø  Sources:
    ·         Text book (Math- Material Addition by Wahyu Tri Astuti.
    ·         Reference book *(Mathematics Revision Guide (Martin Law). * Additional Mathematics ( Ang Tok Woon). * Mathematics IGCSE (Ric Pimentel).
    Ø  Teaching aids:
    ·         LCD
    ·         Note book
    Ø  Individual Task
    Ø  Group Task
    Ø  Examination
    Ø  Quiz
    Ø  Multiple Choice test
    Ø  Matching test
    Ø  Essay test


     
    Materials
    Learning activities
    Time Allotment

    Sources

    Assessment

    Methods
    Instrument
    1
    2
    3
    4
    5
    6
    7
    8
    *      6.2
    To find the distance from point to line and from point to plane in the three dimensional space.
    Þ    To find the distance between  point and line in three dimensional space.
    Þ    To find the distance between point and plane in three dimensional space.
    Þ    To find the distance between two lines in three dimensional space.
     Distance in three dimensional space.    
    Þ    Identifying the mean of distance among point, line and plane in three dimensional space.
    Þ    Calculating the distance between point and the line in three dimensional space.
    Þ    Calculating the point distance and the plane in three dimensional space.
    Þ    Calculating the distance between two lines in three dimensional space.
    ·         10 x 45 minutes
    ·          
    Ø  Sources:
    ·         Text book (Math- Material Addition by Wahyu Tri Astuti.
    ·         Reference book *(Mathematics Revision Guide (Martin Law). * Additional Mathematics ( Ang Tok Woon). * Mathematics IGCSE (Ric Pimentel).
    Ø  Teaching aids:
    ·         LCD
    ·         Note book
    Ø  Individual Task
    Ø  Group Task
    Ø  Examination
    Ø  Quiz
    Ø  Multiple Choice test
    Ø  Matching test
    Ø  Essay test



























    Basic Competency

    Indicator
    Materials
    Learning activities
    Time Allotment

    Sources

    Assessment

    Methods
    Instrument
    1
    2
    3
    4
    5
    6
    7
    8
    *      6.3
    To find the size of angle between the line and plane and between two planes in the three dimensional space.
    Þ    To find the measure of  angle between two lines in three dimensional space.
    Þ    To find the angle size between the line and the plane in three dimensional space.
    Þ    To find the angle size between two planes in three dimensional space.
    Angle in three dimensional space.   
    Þ    Identifying the meaning  of angle between the point, line and plane in three dimensional space.
    Þ    Drawing the angle between two lines in three dimensional space.
    Þ    Calculating the measure of angle between two lines in the three dimensional space.
    Þ    Drawing the angle between the line and the plane in three dimensional space.
    Þ    Calculating the angle between the line and the plane in three dimensional space.
    Þ    Drawing the angle between two planes   the three dimensional space.
    Þ    Calculating the angle size between two planes in three dimensional space





    ·         20 x 45 minutes



    ·          
    Ø  Sources:
    ·         Text book (Math- Material Addition by Wahyu Tri Astuti.
    ·         Reference book *(Mathematics Revision Guide (Martin Law). * Additional Mathematics ( Ang Tok Woon). * Mathematics IGCSE (Ric Pimentel).
    Ø  Teaching aids:
    ·         LCD
    ·         Note book
    Ø  Individual Task
    Ø  Group Task
    Ø  Examination
    Ø  Quiz
    Ø  Multiple Choice test
    Ø  Matching test
    Ø  Essay test

    Approved by
    The Principal of SMA N 1 Salatiga


    Drs. SAPTONO N, M.Si, M.Pd

    NIP. 19680921 199003 1 006

    Salatiga, ……………., 2010
    Math Teacher


    Dra. WAHYU TRI ASTUTI, M. Pd

    NIP. 19670908 199802 2 004

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