SOLUTION: how to find the value of y? 4^y+4^(2*y+1)=1

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Question 657728: how to find the value of y?
4^y+4^(2*y+1)=1

Answer by chriswen(106) About Me  (Show Source):
You can put this solution on YOUR website!
4^y+4^(2*y+1)=1
Simplify the question by making a substitution.
Let x = 4^y
4%5Ey+%2B+4%5E%282y%2B1%29=1
4%5Ey+%2B+4%284%5Ey%29%5E2=1
4x%5E2+%2B+x+=+1
4x%5E2+%2B+x+-+1+=+0
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 4x%5E2%2B1x%2B-1+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%281%29%5E2-4%2A4%2A-1=17.

Discriminant d=17 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-1%2B-sqrt%28+17+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%281%29%2Bsqrt%28+17+%29%29%2F2%5C4+=+0.390388203202208
x%5B2%5D+=+%28-%281%29-sqrt%28+17+%29%29%2F2%5C4+=+-0.640388203202208

Quadratic expression 4x%5E2%2B1x%2B-1 can be factored:
4x%5E2%2B1x%2B-1+=+4%28x-0.390388203202208%29%2A%28x--0.640388203202208%29
Again, the answer is: 0.390388203202208, -0.640388203202208. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+4%2Ax%5E2%2B1%2Ax%2B-1+%29


0.390388203202208, -0.640388203202208
sub into x = 4^y
logx = ylog4
y = logx/log4
y1 = -0.6785
y2 = invalid (can't log negative numbers)
The answer is approximately -0.6785. If you want exact formula you'll need to use the quadratic formula to solve and leave in exact numbers.