9x(x+6)-(3x+1)²=1

Let's solve the equation step by step:

Given equation: \[ 9x(x + 6) - (3x + 1)^2 = 1 \]

1. Expand the terms on the left side:

   • Start with \( 9x(x + 6) \): \[ 9x^2 + 54x \]

   • Now expand \( (3x + 1)^2 \): \[ (3x + 1)(3x + 1) = 9x^2 + 6x + 1 \]

2. Substitute these expansions back into the equation: \[ 9x^2 + 54x - (9x^2 + 6x + 1) = 1 \]

3. Simplify the equation:

   • Distributing the negative sign: \[ 9x^2 + 54x - 9x^2 - 6x - 1 = 1 \]

   • Combine like terms:

\[ (54x - 6x) - 1 = 1 \] \[ 48x - 1 = 1 \]

4. Add 1 to both sides: \[ 48x = 2 \]

5. Divide by 48: \[ x = \frac{2}{48} = \frac{1}{24} \]

Thus, the solution to the equation is: \[ \boxed{\frac{1}{24}} \]