Solve the quadratic equations by factorization method only x2 + x + 1 = 0

Given: x² + x + 1 = 0

\( x^2 + x + \dfrac{1}{4} + \dfrac{3}{4} = 0\)

x² + 2 (x) (\( \dfrac{1}{2}\)) + (\( \dfrac{1}{2}\))² + \( \dfrac{3}{4}\) = 0

(x + \( \dfrac{1}{2}\))² + \( \dfrac{3}{4}\) = 0 [Since, (a + b)² = a² + 2ab + b²]

(x + \( \dfrac{1}{2}\))² + \( \dfrac{3}{4}\) × 1 = 0

We know, i² = –1 = 1 = –i²

By substituting 1 = –i² in the above equation, we get

(x + \( \dfrac{1}{2}\))² + \( \dfrac{3}{4}\) (-1)² = 0

(x + \( \dfrac{1}{2}\))² +  \( \dfrac{3}{4}\)i² = 0

(x + \( \dfrac{1}{2}\))² – \( \sqrt{(\dfrac{3i}{2})^2}=0\)

[By using the formula, a² – b² = (a + b) (a – b)]

\( (x + \dfrac{1}{2} + \sqrt{(\dfrac{3i}{2}})) (x + \dfrac{1}{2} – \sqrt{(\dfrac{3i}{2}})) = 0\)

The roots of the given equation are \( \dfrac{-1}{2}\) + \( \sqrt{(\dfrac{3i}{2})}\), \( \dfrac{-1}{2}\) – \( \sqrt{(\dfrac{3i}{2})^2}\)