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!
(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
Yes, both of your answers are correct!
For the first question, the equation x^2 + 22x + k = (x-r)^2 holds true for all values of x. To find the sum of r and k, we can compare the coefficients of x on both sides of the equation. On the left side, the coefficient of x is 22, and on the right side, it is 2r. Therefore, we can equate these two expressions:
22 = 2r
Solving for r gives us r = 11.
Now, substituting r = 11 back into the equation, we have:
x^2 + 22x + k = (x-11)^2
Expanding the right side of the equation, we get:
x^2 + 22x + k = x^2 - 22x + 121
Comparing the coefficients of x again, we see that k = 121.
The sum of r and k is therefore:
r + k = 11 + 121 = 132.
So, your answer of 110 is incorrect. The correct answer is 132.
For the second question, you correctly expanded the expression (a - 7)(a + j) and set it equal to a^2 + 2a + k. Collecting like terms, we have:
a^2 + aj - 7a - 7j = a^2 + 2a + k
Comparing the coefficients of a on both sides, we see that aj - 7a = 2a. Simplifying this equation, we get:
aj - 9a = 0
Factoring out an a on the left side, we have:
a(j - 9) = 0
Since this equation holds true for all values of a, j - 9 = 0. Solving for j, we get j = 9.
Substituting j = 9 into the original equation, we have:
(a - 7)(a + 9) = a^2 + 2a + k
Expanding the left side gives us:
a^2 + 9a - 7a - 63 = a^2 + 2a + k
Combining like terms, we find that k = -63.
So, your answer of -63 is correct. Well done!