SOLUTION: The quadratic function {{{f(x)=ax^2+bx+c}}} has the following characteristics: (i) passes through the point (2,4); (ii) has a maximum value of 6 when x=4; and (iii) has a zero of {

Question 979388: The quadratic function has the following characteristics: (i) passes through the point (2,4); (ii) has a maximum value of 6 when x=4; and (iii) has a zero of .
Find the values of a,b, and c.
Found 2 solutions by josgarithmetic, Boreal:
Answer by josgarithmetic(39617) (Show Source): You can put this solution on YOUR website!
The maximum is a vertex, (4,6) and you can say .

The given zero should be enough for finding value of a. Expect to use or find .

Standard Form function is .
Simply do the simplification (multiplication...) to put into the general form to identify b and c.
You already know a.
Answer by Boreal(15235) (Show Source): You can put this solution on YOUR website!
f(4)=6
If there is a zero at 4+2 sqrt (3), there is a zero at 4 - 2 sqrt (3)
a*4+b*2+c=4
4a+2b+c=4
16a+4b+c=6
Eliminate c
4a+2b+c=4
-16a-4b-c=-6
-12a-2b=-2
12a+2b=2
6a+b=1
But -b/2a=4, so -b=8a
-2a=1
a=(-1/2)
b=4
first equation -2+8+c=4; c=-2
second equation -8+16+c=6; c=-2
(-1/2)x^2+4x-2=f(x)
quadratic formula:
{-4 +/- sqrt (16-4)}/-1
roots are 4+/- sqrt (12); sqrt (12=2 sqrt (3))
a= -1/2
b=4
c= -2

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