2x^2 + 2x + 1 = 0

Steps to Solve

1. Identify coefficients Identify the coefficients in the quadratic equation (2x^2 + 2x + 1 = 0). Here, (a = 2), (b = 2), and (c = 1).

2. Apply the quadratic formula Use the quadratic formula to find the values of (x): $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ Substituting the values of (a), (b), and (c): $$ x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 2 \cdot 1}}{2 \cdot 2} $$

3. Calculate the discriminant Calculate the value under the square root (the discriminant): $$ b^2 - 4ac = 2^2 - 4 \cdot 2 \cdot 1 = 4 - 8 = -4 $$

4. Interpret the discriminant Since the discriminant is negative ((-4)), this indicates that there are no real solutions. Instead, we will have complex solutions.

5. Find the complex solutions Now substitute back this value into the quadratic formula: $$ x = \frac{-2 \pm \sqrt{-4}}{4} $$ Recall that ( \sqrt{-4} = 2i), where (i) is the imaginary unit.

So we have: $$ x = \frac{-2 \pm 2i}{4} = \frac{-1 \pm i}{2} $$

Thus the solutions are: $$ x = -\frac{1}{2} + \frac{i}{2} \quad \text{and} \quad x = -\frac{1}{2} - \frac{i}{2} $$