Solve for x e^(2x)-e^x-12=0

Problem

e(2*x)−ex−12=0

Solution

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

2. Rewrite the equation using the substitution, noting that e(2*x)=(ex)2=u2

u2−u−12=0

3. Factor the quadratic equation by finding two numbers that multiply to −12 and add to −1

(u−4)*(u+3)=0

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

u=4

u=−3

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

ex=4

ex=−3

6. Evaluate the solutions. For ex=4 take the natural logarithm of both sides.

x=ln(4)

7. Discard the extraneous solution. Since ex must always be greater than 0 the equation ex=−3 has no real solution.

Final Answer

x=ln(4)

Want more problems? Check here!