Solve the Quadratic Equation: -81x² + 54x - 9 = 0

To solve this problem, we'll follow these steps:

• Step 1: Identify the coefficients $a = -81$, $b = 54$, and $c = -9$.
• Step 2: Calculate the discriminant $D = b^2 - 4ac$.
• Step 3: Apply the quadratic formula to find the solution for $x$.

Step 1: We have $a = -81$, $b = 54$, and $c = -9$.

Step 2: Calculate the discriminant:

$D = b^2 - 4ac = 54^2 - 4(-81)(-9)$

$= 2916 - 2916 = 0$

Since the discriminant is zero, there is exactly one real solution, indicating a perfect square trinomial.

Step 3: Apply the quadratic formula:

$x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-54 \pm \sqrt{0}}{-162}$

$x = \frac{-54}{-162} = \frac{1}{3}$

Therefore, the solution to the problem is $x = \frac{1}{3}$.