Solve for x e^(2x)-10e^x+9=0

Problem

e(2*x)−10*ex+9=0

Solution

1. Identify the equation as a quadratic in form by recognizing that e(2*x)=(ex)2

2. Substitute a new variable u=ex to rewrite the equation as a standard quadratic.

u2−10*u+9=0

3. Factor the quadratic equation by finding two numbers that multiply to 9 and add to −10

(u−9)*(u−1)=0

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

u=9

u=1

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

ex=9

ex=1

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

x=ln(9)

x=ln(1)

7. Simplify the results using logarithmic properties, noting that ln(1)=0 and ln(9)=ln(3)=2*ln(3)

x=2*ln(3)

x=0

Final Answer

x=0,2*ln(3)

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