Solve for x 3^(2x)-12*3^x+27=0

Problem

3(2*x)−12⋅3+27=0

Solution

1. Rewrite the first term using the power of a power rule, a(b*c)=(ab)c to express the equation in terms of 3

(3)2−12⋅3+27=0

2. Substitute a new variable u=3 to transform the equation into a standard quadratic form.

u2−12*u+27=0

3. Factor the quadratic equation by finding two numbers that multiply to 27 and add to −12

(u−9)*(u−3)=0

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

u=9

u=3

5. Back-substitute 3 for u to solve for the original variable x

3=9⇒3=3⇒x=2

3=3⇒3=3⇒x=1

Final Answer

x=1,2

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