Using the given three equations F(x) = x² - 8x + 12, G(x) = 20 - 4x, and H(x) = (x - 6)², we get:
a) H(10) = 16,
b) F(2) = 0,
c) G(3) + H(4) = 12,
d) H(x) = x² - 12x + 36, and
e) F(x) + H(x) = 2x² - 20x + 48.
In the question, we are given three equations:
F(x) = x² - 8x + 12,
G(x) = 20 - 4x, and
H(x) = (x - 6)².
a) We are asked to determine H(10).
To do this, we substitute x = 10 in H(x), to get:
H(x) = (x - 6)²,
or, H(10) = (10 - 6)²,
or, H(10) = 4²,
or, H(10) = 16.
Thus, H(10) = 16.
b) We are asked to determine F(2).
To do this, we substitute x = 2 in F(x), to get:
F(x) = x² - 8x + 12,
or, F(2) = 2² - 8(2) + 12,
or, F(2) = 4 - 16 + 12,
or, F(2) = 0.
Thus, F(2) = 0.
c) We are asked to determine G(3) + H(4).
First we determine G(3) by substituting x = 3 in G(x), to get:
G(x) = 20 - 4x,
or, G(3) = 20 - 4(3),
or, G(3) = 20 - 12,
or, G(3) = 8.
Now, we determine H(4) by substituting x = 4 in H(x), to get:
H(x) = (x - 6)²,
or, H(4) = (4 - 6)²,
or, H(4) = (-2)²,
or, H(4) = 4.
Now, to determine G(3) + H(4), we substitute G(3) = 8, and H(4) = 4, to get:
G(3) + H(4)
= 8 + 4
= 12.
Thus, G(3) + H(4) = 12.
d) We are asked to simplify H(x) so that it does not have any parenthesis.
Now, H(x) = (x - 6)² is in the form of (a - b)².
We know that the simplification of the form (a - b)² is a² - 2ab + b².
Using the same, we get:
H(x) = (x - 6)²,
or, H(x) = x² - 2(x)(6) + 6²,
or, H(x) = x² - 12x + 36.
Thus, H(x) = (x - 6)², on simplifying to remove the parenthesis looks like, H(x) = x² - 12x + 36.
e) We are asked to determine F(x) + H(x).
We know that F(x) = x² - 8x + 12 and H(x) = x² - 12x + 36.
Adding the two, we get:
F(x) + H(x) = (x² - 8x + 12) + (x² - 12x + 36),
or, F(x) + H(x) = 2x² - 20x + 48.
Thus, F(x) + H(x) = 2x² - 20x + 48.
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