Solve for x 3^(x-x^2)=1/(9^x)

Problem

3^(x−x^2)=1/9^x

Solution

1. Rewrite the right side using a common base of 3 by noting that 9=3^2

3^(x−x^2)=1/((3^2)^x)

2. Apply exponent rules to simplify the right side, specifically (a^m)^n=a^(m*n) and 1/(a^n)=a^(−n)

3^(x−x^2)=3^(−2*x)

3. Equate the exponents since the bases are equal and the function ƒ(x)=3^x is one-to-one.

x−x^2=−2*x

4. Rearrange the equation into a standard quadratic form by moving all terms to one side.

x−x^2+2*x=0

−x^2+3*x=0

5. Factor the quadratic expression to find the roots.

−x*(x−3)=0

6. Solve for x by setting each factor equal to zero.

x=0

x−3=0⇒x=3

Final Answer

x=0,3

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