Solve the following quadratic equation using the quadratic formula. \[ -6 x^{2}+3=4 x \]

We start with the equation \[ -6x^2 + 3 = 4x. \] **Step 1.** Rewrite the equation in standard form. Subtract \(4x\) from both sides to obtain \[ -6x^2 - 4x + 3 = 0. \] **Step 2.** Identify the coefficients. In the standard quadratic form \(ax^2 + bx + c = 0\), \[ a = -6,\quad b = -4,\quad c = 3. \] **Step 3.** Write down the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \] **Step 4.** Substitute the coefficients into the formula: \[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(-6)(3)}}{2(-6)}. \] Simplify the expression step by step: - Compute \(-b\): \[ -(-4) = 4. \] - Compute \(b^2\): \[ (-4)^2 = 16. \] - Compute \(-4ac\): \[ -4(-6)(3) = 72. \] - The discriminant becomes: \[ b^2 - 4ac = 16 + 72 = 88. \] Now we have: \[ x = \frac{4 \pm \sqrt{88}}{-12}. \] **Step 5.** Simplify the square root: \[ \sqrt{88} = \sqrt{4 \times 22} = 2\sqrt{22}. \] Thus, \[ x = \frac{4 \pm 2\sqrt{22}}{-12}. \] **Step 6.** Factor the numerator: \[ x = \frac{2(2 \pm \sqrt{22})}{-12}. \] Reduce the fraction by dividing both the numerator and the denominator by \(2\): \[ x = \frac{2 \pm \sqrt{22}}{-6}. \] **Step 7.** Multiply numerator and denominator by \(-1\) to write the solutions in a standard form: \[ x = -\frac{2 \pm \sqrt{22}}{6}. \] This can be expressed as two separate solutions: \[ x = \frac{-2 + \sqrt{22}}{6} \quad \text{or} \quad x = \frac{-2 - \sqrt{22}}{6}. \]