4x2-6x+4=0

To solve the quadratic equation \( 4x^2 - 6x + 4 = 0 \), we can use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \( a = 4 \), \( b = -6 \), and \( c = 4 \).

Now, we can calculate the discriminant \( b^2 - 4ac \):

\[ b^2 - 4ac = (-6)^2 - 4 \cdot 4 \cdot 4 = 36 - 64 = -28 \]

Since the discriminant is negative (\(-28\)), this indicates that there are no real solutions to the equation. Instead, there will be two complex solutions.

Now we can substitute the values into the quadratic formula:

\[ x = \frac{-(-6) \pm \sqrt{-28}}{2 \cdot 4} = \frac{6 \pm \sqrt{-28}}{8} \]

We can simplify \( \sqrt{-28} \):

\[ \sqrt{-28} = \sqrt{28}i = \sqrt{4 \cdot 7}i = 2\sqrt{7}i \]

Now substituting this back into our formula:

\[ x = \frac{6 \pm 2\sqrt{7}i}{8} = \frac{6}{8} \pm \frac{2\sqrt{7}i}{8} = \frac{3}{4} \pm \frac{\sqrt{7}}{4}i \]

Thus, the solutions to the equation \( 4x^2 - 6x + 4 = 0 \) are:

\[ x = \frac{3}{4} + \frac{\sqrt{7}}{4}i \quad \text{and} \quad x = \frac{3}{4} - \frac{\sqrt{7}}{4}i \]