\( 1.1 .1 \quad x(6 x-5)=6 \)

We begin with the equation \[ x(6x-5) = 6. \] **Step 1. Expand the equation.** Multiply out the left-hand side: \[ 6x^2 - 5x = 6. \] **Step 2. Rearrange the equation to set it equal to zero.** Subtract \(6\) from both sides: \[ 6x^2 - 5x - 6 = 0. \] **Step 3. Identify the coefficients for the quadratic equation.** The quadratic equation is of the form \(ax^2 + bx + c = 0\), where: \[ a = 6,\quad b = -5,\quad c = -6. \] **Step 4. Compute the discriminant \(\Delta\).** The discriminant is given by: \[ \Delta = b^2 - 4ac. \] Substitute the values: \[ \Delta = (-5)^2 - 4(6)(-6) = 25 + 144 = 169. \] **Step 5. Solve using the quadratic formula.** The quadratic formula is: \[ x = \frac{-b \pm \sqrt{\Delta}}{2a}. \] Substitute the known values: \[ x = \frac{-(-5) \pm \sqrt{169}}{2 \cdot 6} = \frac{5 \pm 13}{12}. \] **Step 6. Determine the two solutions.** 1. When using the positive square root: \[ x = \frac{5 + 13}{12} = \frac{18}{12} = \frac{3}{2}. \] 2. When using the negative square root: \[ x = \frac{5 - 13}{12} = \frac{-8}{12} = -\frac{2}{3}. \] **Final Answer:** \[ x = \frac{3}{2} \quad \text{or} \quad x = -\frac{2}{3}. \]