Answer: k = -15
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Explanation:
Let p(x) = 4x^2+4x+k be the polynomial function.
Also, let r and s be the two roots of the polynomial p(x).
By definition of what it means to be a root, we know that
p(r) = 0
p(s) = 0
So this means p(r) = p(s).
Because one root exceeds another by 4, we can say s = r+4.
So the equation p(r) = p(s) updates to p(r) = p(r+4).
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Let's compute p(r) and p(r+4)
So,
p(x) = 4x^2+4x+k
p(r) = 4r^2+4r+k
and
p(x) = 4x^2+4x+k
p(r+4) = 4(r+4)^2+4(r+4)+k
p(r+4) = 4(r^2+8r+16)+4(r+4)+k
p(r+4) = 4r^2+32r+64+4r+16+k
p(r+4) = 4r^2+36r+80+k
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Now equate those results
p(r) = p(r+4)
4r^2+4r+k = 4r^2+36r+80+k
4r+k = 36r+80+k ...... the 4r^2 terms cancel
4r = 36r+80 ..... the k terms cancel as well
4r-36r = 80
-32r = 80
r = 80/(-32)
r = (16*5)/(-16*2)
r = -5/2 = -2.5 is one of the roots
s = r+4
s = -2.5+4
s = 1.5 = 3/2 is the other root.
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With this in mind, we can use either r or s to find the value of k
p(x) = 4x^2 + 4x + k
p(r) = 4r^2 + 4r + k
p(r) = 4(-2.5)^2 + 4(-2.5) + k
p(r) = 15+k
0 = 15+k
k+15 = 0
k = -15
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To confirm this answer, you can use the quadratic formula to solve 4x^2+4x-15 = 0. You should get the two roots r = -5/2 = -2.5 and s = 3/2 = 1.5
Then note how s-r = 4 which is the same as saying s = r+4.
