Step 1 :First, rewrite the equation in standard form: \(4x^2 - 27x - 81 = 0\).
Step 2 :Next, factor the quadratic equation. To do this, we need to find two numbers that multiply to \(-4 \times -81 = 324\) and add to \(-27\). These numbers are \(-18\) and \(-9\).
Step 3 :So, we can rewrite the equation as \((4x^2 - 18x) - (9x + 81) = 0\).
Step 4 :Factor by grouping: \(2x(2x - 9) - 9(2x - 9) = 0\).
Step 5 :This gives us \((2x - 9)(2x - 9) = 0\).
Step 6 :Setting each factor equal to zero gives the solutions \(x = \frac{9}{2}\) and \(x = \frac{9}{2}\).
Step 7 :So, the solution set is \(\boxed{\{\frac{9}{2}\}}\).
Step 8 :Finally, we can check our solution by substituting \(x = \frac{9}{2}\) back into the original equation: \(4(\frac{9}{2})^2 = 27(\frac{9}{2}) + 81\), which simplifies to \(81 = 81\), confirming that our solution is correct.