Solve for x 6e^(2x)+5e^x-4=0

Problem

6*e(2*x)+5*ex−4=0

Solution

1. Substitute a new variable to transform the equation into a quadratic form by letting u=ex

6*u2+5*u−4=0

2. Factor the quadratic equation by finding two numbers that multiply to 6⋅(−4)=−24 and add to 5 These numbers are 8 and −3

6*u2+8*u−3*u−4=0

2*u*(3*u+4)−1*(3*u+4)=0

(2*u−1)*(3*u+4)=0

3. Solve for u by setting each factor to zero.

2*u−1=0⇒u=1/2

3*u+4=0⇒u=−4/3

4. Back-substitute ex for u to solve for x

ex=1/2

ex=−4/3

5. Apply the natural logarithm to solve for x Note that ex must be positive, so ex=−4/3 has no real solution.

x=ln(1/2)

6. Simplify the expression using logarithm properties.

x=−ln(2)

Final Answer

x=−ln(2)

Want more problems? Check here!