6(4x - 3) - 4x² + 3x = 0

To solve the equation \(6(4x - 3) - 4x^2 + 3x = 0\), we will first simplify it step by step:

1. Distribute \(6\) in the first term:
\[
6(4x - 3) = 24x - 18
\]

2. Substitute this into the equation:
\[
24x - 18 - 4x^2 + 3x = 0
\]

3. Combine like terms (\(24x + 3x = 27x\)):
\[
-4x^2 + 27x - 18 = 0
\]

4. Rearranging gives:
\[
4x^2 - 27x + 18 = 0
\]

Now we will use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to solve for \(x\). Here, \(a = 4\), \(b = -27\), and \(c = 18\).

1. Calculate the discriminant:
\[
b^2 - 4ac = (-27)^2 - 4(4)(18) = 729 - 288 = 441
\]

2. Calculate the roots:
\[
x = \frac{27 \pm \sqrt{441}}{2 \cdot 4} = \frac{27 \pm 21}{8}
\]

This results in two possible values for \(x\):
1. \(x = \frac{27 + 21}{8} = \frac{48}{8} = 6\)
2. \(x = \frac{27 - 21}{8} = \frac{6}{8} = \frac{3}{4}\)

**Final Solutions:**
\[
x = 6 \quad \text{or} \quad x = \frac{3}{4}
\]