4х²=2x-3

To solve the equation \(4x^2 = 2x - 3\), we first rearrange it into standard quadratic form, \(ax^2 + bx + c = 0\).

1. Move all terms to one side of the equation: \[ 4x^2 - 2x + 3 = 0 \]

2. Now, we can identify the coefficients for the quadratic equation:

   • \(a = 4\)
   • \(b = -2\)
   • \(c = 3\)

3. Next, we can use the quadratic formula: \[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \]

4. We calculate the discriminant (\(b^2 - 4ac\)): \[ b^2 - 4ac = (-2)^2 - 4(4)(3) = 4 - 48 = -44 \]

Since the discriminant is negative (\(-44\)), the equation has no real solutions, but it has two complex solutions.

5. We can find these complex solutions: \[ x = \frac{{-(-2) \pm \sqrt{{-44}}}}{2 \cdot 4} = \frac{{2 \pm \sqrt{{-44}}}}{8} = \frac{{2 \pm \sqrt{44}i}}{8} \] \[ \sqrt{44} = \sqrt{4 \cdot 11} = 2\sqrt{11} \] Therefore: \[ x = \frac{{2 \pm 2\sqrt{11}i}}{8} = \frac{{1 \pm \sqrt{11}i}}{4} \]

The final solutions are: \[ x = \frac{1 + \sqrt{11}i}{4} \quad \text{and} \quad x = \frac{1 - \sqrt{11}i}{4} \]