a) log x (x + 6) = 2
Means that
x^2 = x + 6
x^2 - x - 6 = 0
(x - 3) ( x+ 2) = 0
x - 3 = 0 x + 2 = 0
x = 3 x = - 2 ( reject )
b) log 2 (3x +2) = 5
2^5 = 3x + 2
32 = 3x + 2
32 - 2 = 3x
30 = 3x
10 = x
c) log 2 ( 3x - 3) = log 2 9
Implies that
3x - 3 = 9
3x = 9 + 3
3x = 12
x = 4
d) log ( x + 4) + log )x - 4) = 2 log 3
We can write
log [ ( x + 4) (x - 4) ] = log 3^2
So
(x + 4) ( x - 4) = 3^2
x^2 - 16 = 9
x^2 = 16 + 9
x^2 = 25 take the positive root
x = 5
3) (x^4 + 4x^3 - 5x - 20) / (x + 4)
We are testing to see if - 4 is a root....so....
-4 [ 1 4 0 - 5 -20 ]
-4 0 0 20
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1 0 0 -5 0
-4 is a root
The remaining polynomial is x^3 - 5
2) 4x^2 - 6x - 11
p = ± [ 1 , 11]
q = ± [ 1 , 2 , 4 ]
p/q = ± [ 1 , 1/2 1/4 , 11 , 11/2 , 11/4 ]
1)
- 3 [ 1 - 1 0 - 3 - 2 ]
- 3 12 -36 117
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1 -4 12 -39 115 not a root
-1 [ 1 - 1 0 - 3 -2 ]
-1 2 - 2 5
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1 -2 2 -5 3 not a root
2 [ 1 - 1 0 - 3 - 2 ]
2 2 4 2
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1 1 2 1 0 is a root
The area is enlarged by 3^2 = 9 times
To see this....let the original area = L * W
Then the new area = (3L) * (3W) = 9 (L*W) = 9 times the original area
AQ * BQ = CQ * DQ
6 * 12 = x * (38 - x)
72 = 38x - x^2 rearrange as
x^2 - 38x + 72 = 0 factor
(x - 36) ( x - 2) = 0
The second factor set to 0 and solved for x gives the minimum length of CQ
x - 2 = 0
x = 2 = CQ
Welcome !!!!
x^2 + 7x + 13
x^2 + 4 [ x^4 + 7x^3 + 17x^2 + 20x + 0 ]
x^4 + 4x^2
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7x^3 + 13x^2 + 20x
7x^3 + 28x
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13x^2 - 8x + 0
13x^2 + 52
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-8x - 52
x^2 + 7x + 13 R - (8x + 52) / (x^2 + 4)
Since g(z) is of a lesser degree than f(z)....then f(z) + g(z) will also be degree four
y = x^2 - 7
The minimum value of x^2 = 0
y = 0 - 7 = - 7 = the minimum value of y
