show all steps solve 6e^x-4e^-x=5 for x.
To solve the equation 6e^x - 4e^-x = 5 for x, we can follow these steps:
Step 1: Start with the given equation: 6e^x - 4e^-x = 5.
Step 2: Multiply the entire equation by e^x to eliminate the negative exponent:
6e^x * e^x - 4e^-x * e^x = 5 * e^x.
Simplifying this equation, we get:
6e^(2x) - 4 = 5e^x.
Step 3: Move all terms to one side of the equation to obtain a quadratic equation:
6e^(2x) - 5e^x - 4 = 0.
Step 4: Make a substitution to simplify the equation further. Let y = e^x.
Substituting this in, the equation becomes:
6y^2 - 5y - 4 = 0.
Step 5: Solve the quadratic equation for y by factoring, completing the square, or using the quadratic formula. In this case, the equation does not factor easily, so we can use the quadratic formula:
The quadratic formula is given by: y = (-b ± √(b^2 - 4ac)) / (2a).
The coefficients of our equation are: a = 6, b = -5, c = -4.
Using the quadratic formula, we get:
y = (-(-5) ± √((-5)^2 - 4 * 6 * (-4))) / (2 * 6).
y = (5 ± √(25 + 96)) / 12.
y = (5 ± √121) / 12.
Simplifying this further, we get two solutions for y:
y = (5 ± 11) / 12.
So, y = (5 + 11) / 12 or y = (5 - 11) / 12.
Simplifying each solution, we get:
y = 16 / 12 or y = -6 / 12.
Therefore, y = 4/3 or y = -1/2.
Step 6: Substitute the values of y back into the substitution made in Step 4 to find e^x:
For y = 4/3:
e^x = 4/3.
Taking the natural logarithm of both sides:
ln(e^x) = ln(4/3).
x = ln(4/3).
For y = -1/2:
e^x = -1/2.
This equation has no real solutions because e^x is always positive.
Step 7: Finalize the solution:
The solution to the equation 6e^x - 4e^-x = 5 for x is x = ln(4/3).
Note: We ignored the solution y = -1/2 in Step 6 since it did not lead to a real solution for x.