• Home
  • Posts RSS
  • Comments RSS
  • Edit

    • mathematics

    January 21, 2011

    | 0 comments

    A quadratic equation is defined as an equation in which one or more of the terms is squared but raised to no higher power. The general form is ax2 + bx + c = 0, where a, b and c are constants.

    In Algebra and Geometry, we learned how to solve linear - quadratic systems algebraically and graphically. With our new found knowledge of quadratics, we are now ready to attack problems that cannot be solved by factoring, and problems with no real solutions

    Remember that linear-quadratic systems of this type can result in three graphical situations such as:
    The equations will intersect in two locations. Two real solutions. The equations will intersect in one location. One real solution. The equations will not intersect.
    No real solutions.

    Keep these images in mind as we proceed to solve these linear-quadratic systems algebraically.

    Example 1:
    		When we studied these systems in Algebra, we encountered situations that could be solved by factoring, such as this first example.

    Solve this system of equations algebraically:
    y = x2 - x - 6     (quadratic equation in one variable of form y = ax2 + bx + c )
    y = 2x - 2     (linear equation of form y = mx + b)

    Substitute from the linear equation into the quadratic equation and solve.
    y     = x2 - x - 6
     2x - 2 = x2 - x - 6
    2x = x2 - x - 4

    0 = x2 - 3x - 4

    0 =(x - 4)(x + 1)

    x - 4 = 0 x + 1 =0
     x = 4 x = -1

    Find the y-values by substituting each value of x into the linear
    equation.

    y = 2(4) - 2 = 6
    POINT (4,6)

    y = 2(-1) - 2 = -4
    POINT (-1,-4)

    See how to use your
    TI-83+/84+ graphing calculator with quadratic-linear systems.
    Click calculator.

    There are 2 "possible" solutions for the system: (4,6) and (-1,-4)
    Check each in both equations.

    y = x2 - x - 6
     6 = (4)2 - 4 - 6 = 6 checks
     y = 2x - 2
     6 = 2(4) - 2 = 6 checks

    y = x2 - x - 6
    -4 = (-1)2 - (-1) - 6 = -4 checks
    y = 2x - 2
     -4 = 2(-1) - 2 = -4 checks

    Answer:

    {(4, 6), (-1, -4)}

    0 comments:

    Post a Comment