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
]
