Expert: Josh - 10/4/2004

Question
A certain quadratic function has the form f(x)+ ax^2 + bx + c, where a, b, and c not equal to zero.If f(-2)=-18, f(0)=-8, f(b)=c, determine algebraically the values of a,b,and c.The answer is: a=-1,b=3,c=-8, but I can't seem to get it. Please explain.Thank you.

Answer
Hi Jeff,

Let's see...To find the unknown coefficients in the quadratic function, we need to find "a","b" and "c" which satisfy the two constraints
f(-2)=-18 ...[Equation #1] and
f(0)=-8   ...[Equation #2]
simultaneously.

For [#1], substituting x=-2 into f(x)=a*x^2+b*x+c yields -18, we get
    f(-2)=4a-2b+c=-18.  ...[#3]
For [#2], substituting x=0 into f(x)=a*x^2+b*x+c, we get
    f(0)=c=-8.  ...[#4]

Now, all the information is contained in [#3] and [#4], because we have already exploited the facts [#1] and [#2].

We only need to focus on [#3] and [#4].
Looking at [#4], the value of c is already known.
That is, c=-8. Putting this into [#3],
we get 4a-2b-8=-18. Rearranging this equation, we get
4a-2b=-10.

Consider two scenarios here.

SCENARIO 1: A more general perspeactive

At this point, if [#1] and [#2] is all that we know, then, the relationship between "a" and "b" is given by
   2a=b-5 ...[#5]

There is an infinite number of solutions to this problem. They are all equally valid, provided that c=-8 and 2a=b-5.

For instance, if you select a=-1, b=2a+5 must equal 3. Hence, a=-1,b=3,c=-8 is one of the many possible solutions.

SCENARIO 2:
BUT, we were also told that f(b)=c. Do not forget this!
This gives us f(b)=a*b^2+b*b+c=c ...[#6]
Subtracting c from both sides, leaves us with
a*b^2+b^2=0...factorizing this,
(a+1)*b^2=0
[In general, (a+1)=0 and b^2=0 are both feasible]
However, we were told in the beginning that b cannot be zero. This implies that a+1=0 or a=-1.

Using this in [#5] gives us b=3.

LESSON: The more constraints we have to work with, the more specific the form of the answer becomes. In scenario 1, we pretend that only conditions #1 and #2 are given, using two indepedent pieces of information, we get rid of two unknown quantities. To fully determine all three unknowns, we require three pieces of information (see scenario 2 or the original question). By including the third constraint, viz., f(b)=c, we refine our answer and make it more specific. In fact, given three independent equations [here, f(-2)=-18,f(0)=-8 and f(b)=c] we can UNIQUELY determine the answer. In other words, only one set of solution is possible.

Cheers.