1. ## Quadratic Expressions and equations

Given that y = 5 + px - (x^2) = 9 - (x+q)^2 , where q > 0, for all values of x,

a) calculate the values of the constants p and q.

b) state the maximum value of y and the value of x at which it occurs.

c) find the set of value of x for which y is negative.

2. Originally Posted by Ilsa
Given that y = 5 + px - (x^2) = 9 - (x+q)^2 , where q > 0, for all values of x,

a) calculate the values of the constants p and q.

b) state the maximum value of y and the value of x at which it occurs.

c) find the set of value of x for which y is negative.
to a):

According to the text of the question

$\displaystyle -x^2+px+5 = -x^2-2qx-q^2+9$

Compare the co-efficients and solve the system of equations for p and q:

$\displaystyle \left|\begin{array}{rcl}p&=&-2q \\ 5&=&-q^2+9\end{array}\right.$

You should come out with p = -4 and q = 2 (the negative solution of q is not allowed).

Thus the quadratic function has the equation: $\displaystyle y = -x^2-4x+5$

to b): Determine the vertex of the parabola by completing the square.

to c) Determine the zeros of the function and consequently those intervals where y < 0.