5.1  Quadratic Roots

Quadratic Equations 

     All quadratic equations are in the form of  ax2 + bx + c = 0, where a, b, and c are real numbers and "a" does not equal 0.

    An important concept associated with quadratic equations is the zero-product property of real numbers.

           pq = 0, if and only if p = 0  or q = 0  (or both equal 0) 

   Remember... to solve a quadratic equation, you need to set the equation equal to zero.

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 The Quadratic Formula   (Right --->                                             

    If you cannot use any of the advanced factoring methods (chapter 2) to find the roots of quadratic equations, you can use the quadratic formula.

   This method is not always the best method to use to find the solution to a quadratic equation.

   It is better to use the discriminant first to determine if there is a real solution.

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  The Discriminant   (Right --->                                                  

     Looking at the quadratic formula, the expression under the radical sign:  b2 - 4ac

    will be very helpful in determining how many solutions we have or if we won't have a real solution.

     It is called the discriminant.

             If b2 - 4ac > 0, the result is positive, then there are two unequal real roots.

             If b2 - 4ac = 0, the result is zero, then there is a repeated real root.  Double Root.

             If b2 - 4ac < 0, the result is negative, then there are no real roots.

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Example 1

  Determine the number of real roots (solutions) for

       x2 - 3x + 4 = 0           a = 1,  b = -3,  c = 4

        b2 - 4ac

       (-3)2 - 4 (1) (4) = 9 - 16 = -7  the result is less than 0, so there are no real roots.

        The graph of this function would not cross the x-axis.

Example 2 

     x2 - 2x + 1 = 0          a = 1,  b = -2,  c = 1

      b2 - 4ac

     (-2)2 - 4(1)(1) = 4 - 4 = 0   the result is equal to zero, so there is a double root. 

     The graph of this function would touch and then turns the x-axis.

 Example 3 

       x2 - 2x - 3 = 0      a = 1,  b = -2,  c = -3

      b2 - 4ac

     (-2)2 - 4(1)(-3) = 4 + 12 = 16    the result is greater than zero, so there are two distinct solutions.

     The graph of this function will cross the x-axis twice