Solve 6e^x -4e^-x = 5 for X
To solve the equation 6e^x - 4e^(-x) = 5 for x, we can use a substitution. Let's substitute e^x with a variable y.
Let y = e^x
Now, we can rewrite the equation in terms of y:
6y - 4/y = 5
To eliminate the fraction, we can multiply both sides of the equation by y:
6y^2 - 4 = 5y
Rearranging the equation:
6y^2 - 5y - 4 = 0
Now, we have a quadratic equation in terms of y. We can solve this equation by factoring or using the quadratic formula.
Factoring is not straightforward, so let's use the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / (2a)
For this equation, a = 6, b = -5, c = -4. Substituting these values:
y = (-(-5) ± √((-5)^2 - 4(6)(-4))) / (2(6))
y = (5 ± √(25 + 96)) / 12
y = (5 ± √(121)) / 12
y = (5 ± 11) / 12
Now, we have two possible values for y:
y1 = (5 + 11) / 12 = 16 / 12 = 4/3
y2 = (5 - 11) / 12 = -6 / 12 = -1/2
Since y = e^x, we can take the natural logarithm of both sides to find x:
ln(y1) = x
ln(y2) = x
Using a calculator:
ln(4/3) ≈ 0.2877
ln(-1/2) is undefined
Therefore, the solution to the equation 6e^x - 4e^(-x) = 5 is:
x ≈ 0.2877