Solve for x e^(2x)-e^x-30=0

Problem

e(2*x)−ex−30=0

Solution

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

2. Rewrite the equation in terms of u noting that e(2*x)=(ex)2=u2

u2−u−30=0

3. Factor the quadratic equation by finding two numbers that multiply to −30 and add to −1

(u−6)*(u+5)=0

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

u=6

u=−5

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

ex=6

ex=−5

6. Evaluate the solutions. For ex=6 take the natural logarithm of both sides to get x=ln(6) For ex=−5 there is no real solution because ex must always be greater than zero.

Final Answer

x=ln(6)

Want more problems? Check here!