Question

1) x^2 + 22x + k = (x-r)^2

Given constants r and k, the above equation is true for all values of x. What is the value of the sum of r and k?

I'm thinking it's 110, because to get 22x, r has to be -11, and that would mean k = 121.
121 - 11 = 110

2) If (a - 7)(a + j) = a^2 + 2a + k for all values of a, where j and k are certain constants, then what is the value of k?

I'm thinking it's -63, because if we expand the first bit, it gives:
a^2 + aj - 7a -7j = a^2 + 2a + k
aj - 7a = 2a
a(j - 7) = 2a
j - 7 = 2
j = 9
-7 x 9 = -63

Are these correct? Thanks in advance!

Answer 1

(x-r)^2 = x^2 - 2rx + r^2
match the terms up with x^2 + 22x + k
so 22x = -2r and k = r^2
Since it is true for all values of x, let x = 1
then 22 = -2r ---< r = -11
and k = 121

so your first one is right

The 2nd is also correct