Unit 10 Radicals and Quadratic Equations

Document Sample
Unit 10 Radicals and Quadratic Equations Powered By Docstoc
					             Unit 10
Radicals and Quadratic Equations
• Learn the Three
  “Nevers” for Radicals
• Simplify Radical
  Expressions
• Solve Quadratic
  Equations
                Lecture 1
Objectives
• Define a radical
  expression
• Simplify perfect
  square factors under
  the radical
• Multiply radicals
  Properties of Radicals

              Radical
    Index
             y
                 x      Radicand

What number multiplied by itself
“y” times gives “x”?
            Example
Simplify:

  2
      25  5
      49  7
      8  42 4       2 2 2
Simplifying Radical Expressions
1. No perfect squares under the radical

        2
            24          46
         4      6           2 6
   Factor out perfect squares!
        Multiplying Radicals
• Radicals with like indices can be multiplied.

             3
                 5   3
                         7  35
                            3




    3 18  54  9 6  3 6
        Homework Set 10.1
• WS Simplifying
  Radicals 10.1
                Lecture 2
Objectives
• Simplify perfect
                            Do you know
  square factors under       what I am?
  the radical
• Multiply radicals
• Work with fractions    I am a Perfect Square!
  and radicals
Simplifying Radical Expressions
2. No fractions under the radical



                             Convert a
        2           2        fraction under
                            the radical into a
                             fraction of
        3           3        radicals!
Simplifying Radical Expressions
3. No radicals in the denominator


    2    3               6            6
          
         3                 
    3                    9           3
  Multiply numerator and denominator by the
  radical in the denominator!
        Homework Set 10.2
• WS Simplifying
  Radicals 10.2
• Quiz next class
  day
                Lecture 3
Objectives
• Define a quadratic
  equation
• Solve quadratic
  equations by taking
  square roots
          Quadratic Equation
A quadratic equation is like a linear equation
 except that it contains a variable raised to
 the second power.


 Quadratic term       Linear term


         ax  bx  c  0
               2

                            Constant term
Solving a Quadratic Equation
Solving a quadratic equation is much more
complicated than solving a linear equation. Try to
solve the following equation using the properties
of equality:


      x  3x  4  0
         2
Solving a Quadratic Equation
While the properties of equality cannot
solve most quadratic equations, they do
work on some of them:


       2x  8  0
             2
Solving a Quadratic Equation by
         Square Roots
   Any quadratic equation the lacks a linear term
   can be solved by:
1. Solving for the squared variable using the
   properties of equality
2. Taking the square root of both sides of the
   equation to solve for the variable
3. Make sure to include the “plus or minus” with
   the square root step, because the square root of a
   number has two answers.

       ax  bx  c  0
             2
                         Example
Solve:   3x  24  0
           2


               3x  24
                 2


                x 8
                 2


               x  8
                2


                 x  2 2
A few more complicated quadratics seem to
  be all “set-up” for square roots:


             ( x  7)  162


            ( x  7)   16
                         2


                  x  7  4
      x  7  4  3 or 11
        Homework Set 10.3
• WS Quadratic
  Equations
  Square Roots
  10.3
Lecture 4
    Objectives
    • Solve quadratic
      equations by
      factoring
Solving a Quadratic Equation
Many quadratic equations can be solved by
factoring them into a product, and then using
something called the zero-product rule.

             x  3x  0
               2                   Factor out the “x”


            x  x  3  0
                                   Apply the zero-product
                                   rule


             x  0 or  3             Viola!
     The Zero-Product Rule
The only way a set of numbers can multiply
to zero is if one of the numbers is zero.

If a b  0, then a  0 or b  0.
While not all quadratics can factor into
products, the zero-product rule is very
useful to solve those that do.
                                Examples
       ( x  7)  0
Solve:               2

           x  7
    (4 y  3)( y  2)  0
            y  3       or 2
                     4
         z (3z  2)( z  2)  0
           z  0 or  2 or 2
                                3
         Factoring Quadratics
    Today we will focus on two kinds of factoring:

1. Factoring out a monomial.
2. Factoring a trinomial square.



     ax  bx  c  0
           2
     Factoring Out a Monomial
When there is no constant term in a quadratic, you
   can always factor out a common monomial
   factor:


        3x  4 x  0
                2

       x(3 x  4)  0
 Then use the zero-product rule to get the solutions.
        Trinomial Squares
Some trinomial quadratics (has all three terms)
factor into perfect squares. The only way they
do this, however, is if their coefficients have a
special relationship given by these forms:


 a  2ab  b   a  b 
   2                    2                    2



 a  2ab  b   a  b 
    2                    2                   2
        Trinomial Squares
Notice that the first and last terms are perfect
squares, and the linear term is two times the
product of the square roots of the ends. Only
very special quadratic equations can be factored
this way.

a  2ab  b   a  b 
   2                   2                  2



 a  2ab  b   a  b 
    2                   2                  2
                                        Examples
Which of these are trinomial squares?


    x  4x  4  0
         2                                yes
  9 x  24 x  16  0
     2                                    yes
   4x  6x  9  0
       2              no
 25x  70 x  49  0
      2               yes
 Solving Trinomial Squares
Trinomial squares are the easiest of all the
quadratics to solve. Simply factor them
according to their form, and then use the zero-
product rule.


a  2ab  b   a  b 
   2                   2                   2



 a  2ab  b   a  b 
    2                   2                   2
                               Examples
Solve:   9 x  24 x  16  0
           2

            (3x  4)  0
                    2


               x4
                      3
          4 x  12 x  9  0
               2



               (2 x  3)  0
                       2


                  x  3
                         2
        Homework Set 10.4
• WS Quadratics
  10.4
                Lecture 5
Objectives
• Solve quadratic
  equations by
  factoring
     Factoring Quadratics
Today we will focus on another kind of
factoring. Many quadratic polynomials are the
result of multiplying two binomials together:

    x  3 x  2  x 2  5x  6
    x  7  x  3  x 2  4 x  21
    x 1 x  5  x 2  4 x  5
        Factoring Quadratics
  We can reverse this process and factor many
  quadratics where the quadratic coefficient is = 1
  by doing the following:


 x  bx  c   x  m x  n
   2


where b  m  n and c  m n
   All you have to do is find two numbers that
   multiply to “c” and add to “b”!
                         Examples
Factor:

   x  3x  2   x  1 x  2
        2


   y  y  20
        2       y  5 y  4

    r  5r  6   r  3 r  2
        2


  z  3z  70   z  10 z  7 
    2
          Factoring Quadratics
     After these quadratics are factored, use the
     zero-product rule to find the solutions:

x  5 x  24   x  8 x  3
 2


               x  8 or 3
        Homework Set 10.5
• WS Quadratics
  10.5
Lecture 6
    Objectives
    • Solve quadratic
      equations by
      factoring
     Factoring Quadratics
Today we will focus on another kind of
factoring. Many quadratic polynomials are the
result of multiplying two binomials together:

   2 x  3 x  2  2 x 2  7 x  6
  3x  7  2x  3  6 x2  5x  21
   5x 13x  5  15x2  22 x  5
          Factoring Quadratics
     We can reverse this process and factor many
     quadratics where the quadratic coefficient is  1
     by doing the following:
1.   Multiply a x c and write it above b.
2.   Find two numbers “m” and “n” so that
     m x n = a x c and m + n = b.
3.   Rewrite the quadratic but replace the linear term
     with two terms with coefficients “m” and “n”.
4.   Factor by grouping.

            ax  bx  c  0
                 2
Factor:
                  6                   Example
Rewrite the
original
              2x  7 x  3
                2                Multiply a x c and
                                 write above b

equation
replacing b
              m  6 and n  1    Find m x n = a x c
                                 and m + n = b


          2 x  6 x  1x  3
              2                       Group and
                                      factor

           2x  6x    1x  3
              2


            2x  x  3  1 x  3 Use the
                                    Distributive

               2x 1 x  3
                                    Property
         Homework Set 10.6
• WS Quadratics
  10.6
• Quiz next class
  day
                Lecture 7
Objectives
• Review Quadratics
• Learn the Factoring
  Map
            Solving Quadratics
   Quadratic equations contain a variable squared
   and cannot be solved by using the properties of
   equality. To solve a quadratic, several different
   methods need to be used:
1. Take square roots
2. Factor:
   1.   Simple monomial
   2.   Trinomial square
   3.   a=1 (short-cut)
   4.   a1 (4-step)
        Homework Set 10.7
• WS Quadratics
  10.7

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:37
posted:11/23/2011
language:English
pages:45