L6 Quadratic Equations and the Quadratic Formula; Applications

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L6 Quadratic Equations and the Quadratic Formula; Applications Powered By Docstoc
					L6 Quadratic Equations and the Quadratic Formula;
   Applications

Quadratic Equations

  A quadratic equation is an equation of the form
                     ax 2 + bx + c = 0
  where a, b, and c are real numbers and a ≠ 0 .

Solving by Factoring
Zero-Product Property:
If ab = 0 , then

Example: Solve by factoring.
   x 2 − 10 x = −9




    4 x 2 − 12 x + 9 = 0




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The Square Roots Method:

   The solution set to x 2 = p ( p ≥ 0) is the set of all
  square roots of the number p, that is
                          x=± p

Example. Solve by using the Square Roots Method.

    x 2 = 25


    ( x + 2) 2 = 3




Solving Quadratic Equations by Completing the Square:
               ax 2 + bx + c = 0,     a≠0
1. Make sure that a = 1; if not, divide each term by a.
2. Get the constant on the right-hand side of the
    equation.
3. Take 1/2 of the coefficient of x, square it, and add
    this number to both sides of the equation.
4. Factor the left-hand side into a perfect square.
5. Solve for x by using the Square Roots Method.

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Example: Solve by completing the square.
   x2 − 8x + 3 = 0




Example: Factor by using completing the square.

    x2 + 4 x − 3 =




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The Quadratic Formula

  The quadratic equation
                ax 2 + bx + c = 0    a≠0
  has the solutions (roots)
                        −b ± b 2 − 4ac
                     x=
                               2a

The quantity b 2 − 4ac , denoted D, is called the
discriminant.
The equation has: two distinct real roots if D > 0
                     one repeated real root if D = 0
                     no real solutions if D < 0

Note: If a < 0 , multiply both sides of the equation by a
−1 to make use of the quadratic formula easier.

The Half-coefficient Quadratic Formula:
If in the quadratic equation
                ax 2 + bx + c = 0 , a≠0
 b is an even number, the following formula for the roots
may be useful:
                      −b
                                  2
                           ⎛b⎞
                         ± ⎜ ⎟ − ac
                   x=
                       2   ⎝2⎠      .
                            a
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Example: Use the discriminant to determine the number
of real solutions of the quadratic equation
     3x 2 − 6 x + 3 = 0




Example: Use the quadratic formula to solve
   3x + x 2 − 1 = 0




    3 x 2 − 2 x − 10 = 0




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The Quadratic Formula and Factoring:

Theorem:     If x1 and x2 are the roots of the quadratic
equation
                      ax 2 + bx + c = 0 ,
then the quadratic trinomial can be factored as follows:
             ax 2 + bx + c = a ( x − x1 )( x − x2 ) .


Example: Factor by using the quadratic formula
               3
   4 x2 + 5x −
               2




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Applications

Example: Find two consecutive positive integers such
that the sum of their squares is 145.




Geometry: Finding the Dimensions
The height of a triangular sign is equal to its base. The
area of the sign is 20 square feet. Find the base and the
height of the sign.




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Physics
Using the position equation
                   s = −16t 2 + υ0t + s0 ,
find the time when an object hits the ground if it is
dropped from a building at a height of 320 feet.




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Dimensions of a Field
A farmer has 380 feet of fencing to enclose two adjacent
fields. Find the dimensions that would enclose an area of
4000 square feet.




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Constructing a Box
An open box is formed by cutting 1.5 inch squares from
each corner of a rectangular piece of metal whose length
is twice its width and bending up the edges. If the box is
to have a volume of 21 cubic inches, what dimensions
should the piece of metal have?




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