Question

1. $-2x^2 - x - 1 = 0$

Explanation:

Step1: Identify quadratic form

The equation is \(-2x^2 - x - 1 = 0\), a quadratic \(ax^2 + bx + c = 0\) with \(a = -2\), \(b = -1\), \(c = -1\).

Step2: Calculate discriminant

Discriminant \(D = b^2 - 4ac\). Substitute values:
\(D = (-1)^2 - 4(-2)(-1) = 1 - 8 = -7\).

Step3: Analyze discriminant

Since \(D = -7 < 0\), the quadratic has no real roots (only complex roots). For complex roots, use quadratic formula \(x = \frac{-b \pm \sqrt{D}}{2a}\).

Substitute \(a, b, D\):
\(x = \frac{-(-1) \pm \sqrt{-7}}{2(-2)} = \frac{1 \pm i\sqrt{7}}{-4} = -\frac{1}{4} \mp \frac{i\sqrt{7}}{4}\).

Answer:

The equation \(-2x^2 - x - 1 = 0\) has no real solutions. Its complex solutions are \(x = -\frac{1}{4} + \frac{i\sqrt{7}}{4}\) and \(x = -\frac{1}{4} - \frac{i\sqrt{7}}{4}\).