+129896 Here's a graph of this one.........
The solutions (roots) are: x = 0 (as Alan noted) and x ≈ .354336....Notice how the lead term "over-powers" the graph very quickly as x moves away from zero in a positive direction. Also, as x gets more and more negative, the first two terms → 0 and the graph has a limit of 1 on the left hand side.....
+4473 4.466^x -2(2.331)^x + 1 = 0
4.466^x - 4.662^x = -1 -->
-0.196^x = -1 -->
log(-0.196^x) = log(-1) -->
Properties of logs allow: x log(-0.196) = log(-1) -->
x = log(-1) / log(-0.196) = Error
However, when plugging in values such as x = 0.02 into -0.196^x = -1, we obtain a number that is close to -1, -0.9679326095185039.
+33661 If only it were that simple; unfortunately ax - bx doesn't equal (a-b)x in general!
4.466x - 2*2.331x + 1 = 0 has one immediately obvious solution; namely x = 0, because:
4.4660 - 2*2.3310 + 1 = 1 - 2*1 + 1 = 0.
I can't see any way of getting another (real number) solution except by a numerical method. A straightforward, though rather inefficient way is just to rearrange the equation as shown below, guess an initial value and then iterate until the iterates converge. It takes about 100 iterations to get 5 decimal places!
+129896 Here's a graph of this one.........
The solutions (roots) are: x = 0 (as Alan noted) and x ≈ .354336....Notice how the lead term "over-powers" the graph very quickly as x moves away from zero in a positive direction. Also, as x gets more and more negative, the first two terms → 0 and the graph has a limit of 1 on the left hand side.....
