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

Problem

e^(2*x)−e^x−12=0

Solution

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

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

u^2−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 e^x for u to solve for x

e^x=4

e^x=−3

6. Evaluate the solutions. For e^x=4 take the natural logarithm of both sides.

x=ln(4)

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

Final Answer

x=ln(4)

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