Solve for x e^(2x)-3e^x+2=0

Problem

e^(2*x)−3*e^x+2=0

Solution

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

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

u^2−3*u+2=0

3. Factor the quadratic equation by finding two numbers that multiply to 2 and add to −3

(u−1)*(u−2)=0

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

u=1

u=2

5. Back-substitute e^x for u to find the values of x

e^x=1

e^x=2

6. Apply the natural logarithm to both sides of each equation to isolate x

x=ln(1)

x=ln(2)

7. Simplify the first result using the property ln(1)=0

x=0

x=ln(2)

Final Answer

x=0,ln(2)

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