Question 23173
<pre>1/2x<sup>2</sup> + 1/3x - 6 = 4
<font size = 3><b>
1. Divide through by the coefficient of x<sup>2</sup>
   
This is 1/2.  To divide by 1/2 is to multiply through by 2

2(1/2x<sup>2</sup>) + 2(1/3x) - 2(6) = 2(4)

           x<sup>2</sup> + 2/3x - 12 = 8

2. Get rid of the constant term on the left by adding its opposite
   to both sides:

                x<sup>2</sup> + 2/3x = 8 + 12
                x<sup>2</sup> + 2/3x = 20

3. Find the square-completing number by 

   (a) multiplying the coefficient of x by 1/2
    
        (2/3)·(1/2) = 1/3

   (b) squaring this result
         
        (1/3)<sup>2</sup> = 1/9

4. Add the square-completing number to both sides

               x<sup>2</sup> + 2/3x + 1/9 = 20 + 1/9

5. Factor the left side, and if everything is done right,
   it will factor into two equal factors, which is a 
   perfect square:

             (x + 1/3)(x + 1/3) = 20 + 1/9

                     (x + 1/3)<sup>2</sup> = 20

6.  Combine the terms on the right

                     (x + 1/3)<sup>2</sup> = 180/9 + 1/9


                     (x + 1/3)<sup>2</sup> = 181/9 


6. Take the square roots of both sides, remembering to put ± on the
   right:         
                                    _____
                        x + 1/3 = ±V181/9

5. Solve for x:
                                          _____
                              x = =1/3 ± <font face = "symbol">Ö</font>181/9

6. Simplify the radical:

   The numerator, 181, is a prime number, so we leave it under the
   radical.  We take the square root of the denominator, 9, and get 3
                                          ___
                              x = -1/3 ± <font face = "symbol">Ö</font>181/3 

If you like you may write this as a single fraction with denomninator 3
                                         ___
                              x = (-1 ± <font face = "symbol">Ö</font>181)/3 


Edwin
AnlytcPhil@aol.com</pre>