Questions: Find all solutions of the equation and express them in the form (a+bi). [x= 2 x^2-2 x+1=0]

Solution Steps

We start with the quadratic equation given by: \[ 2x^2 - 2x + 1 = 0 \]

To find the solutions for the equation \(ax^2 + bx + c = 0\), we can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] In our case, \(a = 2\), \(b = -2\), and \(c = 1\).

First, we calculate the discriminant \(D\): \[ D = b^2 - 4ac = (-2)^2 - 4 \cdot 2 \cdot 1 = 4 - 8 = -4 \] Since the discriminant is negative, we will have complex solutions.

Now we substitute \(a\), \(b\), and \(D\) into the quadratic formula: \[ x = \frac{-(-2) \pm \sqrt{-4}}{2 \cdot 2} = \frac{2 \pm 2i}{4} \]

We can simplify the expression: \[ x = \frac{2}{4} \pm \frac{2i}{4} = \frac{1}{2} \pm \frac{i}{2} \]

Thus, the solutions of the equation \(2x^2 - 2x + 1 = 0\) are: \[ x = \frac{1}{2} - \frac{i}{2} \quad \text{and} \quad x = \frac{1}{2} + \frac{i}{2} \]

Final Answer