algerbra expressions?

Question

the expression 8x-x^2 can be written in the form p-(x-q)^2 for all values of x.

a) find the value of p and the value of q.



b) The expression 8x-x^2 has a maximum value.
i) Find the maximum value of 8x-x^2

ii) state the value for which this maximum value occurs.

Answer 1

Well, I guess there would be a lot of ways to do this problem, so I'll just tell you what is the easiest for me.

8x - x^2 = p - (x - q)^2
8x - x^2 = p - (x^2 - 2xq + q^2)
8x - x^2 = p - x^2 + 2xq - q^2

Note that if p and q are just constants, then 8x should be equivalent to 2xq since 2xq is the only term on the right which has an "x" term.

8x = 2xq
8 = 2q
4 = q

Now, since there are no constants (non-x terms) on the left, then the constant term on the left should be set to 0. Thus,

0 = p - q^2
0 = p - 4^2
0 = p - 16
16 = p

Thus, 8x - x^2 can be expressed as 16 - (x - 2)^2.

Since the term (x - 2)^2 is a square, then the smallest value it can have would be 0. Since the difference would be largest if the subtrahend is smallest, we have

max value = 16 - 0 = 16.

Where does the maximum value occur? Of course, where x - 2 = 0.

x - 2 = 0
x = 2.

Hope that helps.

Answer 2

a) I get p=16, q=4.

b) [To me it would have been more "orderly" to ask ii) first then i.)]
----i) max value for expression is 16
----ii) expression has a max when x = 4

The underlying strategy is this: The p-q expression can equal the x-expression for all x ONLY IF CORRESPONDING POWERS OF X HAVE THE SAME COEFFICIENTS. So what you do is expand and simplify the p-q expression and set its coefficients equal to the corresponding coefficients (of the same x power) in 8x-x^2.

Moja did this, but got careless plugging in his answer(s) after his second "Thus".

Finding critical value(s) of the 1st derivative of your x-expression provides an answer to ii). Then a second derivative confirms the critical value (single number in this case) will generate a max for the x-expression; so you plug the critical value (x=4) into the expression to get the max for i).