A4 Demonstrate an understanding of the roots of quadratic functions

Gabrielle and Lena

Explanation: In a quadratic equation ( y=ax^2+bx+c ) the 'roots' or 'zeros' of the function represtent the values of x at which y=0, otherwise known as the x-intercepts. A graph can have two, one, or no x-intercepts, depending on the equation:

There are four ways of finding the roots of a quadratic function:

 1. The quadratic formula

 2. Factoring

 3. Completing the square

 4. Graphing

1. In each of these cases (discluding graphing) you must first make the y in the equation equal zero, and you must know the a, b, and c values, and place them all on one side of the equals sign. The quadratic formula has a logical derivation, and when you plug in the a, b, and c values from a particular quadratic equation the formula will tell you what the roots of that quadtratic equation are, whether there be 2 imaginary, 2 identical, or two real. This method always works.

2. When you are factoring to determine the roots, you simply let y equal zero, and solve for x.

ex. 0= x^2 +5x+6 <------- you must determine the two numbers which multiply to six, and add to five.

0= (x+2) (x+3) <--- in this case those two numbers were 2 and 3 , and the factors which you can fill in to get the    above equation. You then reason that if those two things mulitply together to give you zero, one factor must itself equal zero. You can then set each factor equal to zero seperately, and solve for x, and the two values that you find will give you the roots, or zeros of the equation. Some equations do not have factors which can be easily determined, so this method is not always desirable

3. To find the roots by completing the square, you change the equation into transformational form as usual, but once again you let y equal zero, and solve for x. You get your two answers in this case, becuase when you must take the square root of both sides of the equation, you get positive and negative answers, and you determine the roots by solving for x using both the positive and negative values.

4. To find the roots when graphing you simply graph the equation, by first putting into transformational form, and using the h,k values ( the vertex) and the stretch as your guide. Unless you are graphing the equation on a calculator, human error is likely in placement of the lines, and estimation of exactly where the lines cross the x axis is unavoidable. For this reason this method is good for little more than estimation.

Example Problem:

Find the roots of the following quadratic equations any of the above methods:

a.y= x^2 + 18x + 72                          b. y=5x^2 + 16x + 8

solutions:

                                                                                                                                          _____

a. ( solved by factoring)                    b. (solved using the quadratic formula)                ( /             = square root of )

y= x^2 + 18x + 72                            y= 5x^2 + 16x + 8                                                       

0= x^2 + 18x + 72                            0= 5x^2 +16x + 8

0= (x+6) (x+12)                                                     _______________

                                                     x=  -16 (+ or -) / (16^2 - (4)(1)(8) )

                                                                                      (1) (2)

0= x+6      0= x+12                                              _____

x= -6         x= -12                       x= -16 (+ or -) / 224

                                                                                 2

                                                                                  ___

                                                     x=  -16 (+ or -)  4 / 14  

                                                                            2

                                                                       ___                           ___

                                                     x= -8 + 2/ 14           x= -8 - 2/14