Mathematics
posted by Raquel on .
How do i solve the pair of simultaneous equations
y=x^23x12 and y=2x16
and epxress y=x^23x12 in the form y=(xh^2)+k, where h and k are constants
Hence determine the minimum value of the function y=x^23x12
I have a few of these to do, so if someone can just show me how to do this one i can do the rest.

in the first:
notice that one equation is quadratic, the other is linear.
sub the linear into the quadratic
x^2  3x  12 = y = 2x  16
x^2  3x  12 = 2x  16
x^2  5x + 4 = 0
(x1)(x4) = 0
x = 1 or x = 4
sub into the easier of the two, namely into y = 2x  16
if x = 1 then y = 2(1)  16 = 14
if x = 4 then y = 2(4)  16 = 8
so two points of intersection, (1,14) and (4,8)
We were lucky that the quadratic factored, if not you will have to use the quadratic equation formula.
for the second, complete the square
y=x^23x12
y = x^2  3x + 9/4  9/4  12
= (x  3/2)^2  57/4
so the min value of the function is 57/4 and it occurs when x = 3/2
or
the vertex is (3/2,57/4) 
Here a link to the graph of this problem:
http://i263.photobucket.com/albums/ii157/mathmate/raquel.png