Solve the Fractional Quadratic: x²/2 + x/4 + 1 = 0

Solve the following equation:

$\frac{x^2}{2}+\frac{x}{4}+1=0$

To solve the quadratic equation $\frac{x^2}{2} + \frac{x}{4} + 1 = 0$, follow these steps:

• Step 1: Convert to Standard Form
  Begin by eliminating the fractions to simplify. Multiply the entire equation by 4, the least common multiple of the denominators: $4 \left( \frac{x^2}{2} + \frac{x}{4} + 1 \right) = 4 \cdot 0$ which simplifies to: $2x^2 + x + 4 = 0$ Now, it's in standard quadratic form: $ax^2 + bx + c = 0$, with $a = 2$, $b = 1$, and $c = 4$.
• Step 2: Evaluate the Discriminant
  The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is calculated as: $b^2 - 4ac$ Substituting the values, we have: $1^2 - 4 \cdot 2 \cdot 4 = 1 - 32 = -31$ Since the discriminant is negative, it indicates that there are no real solutions for the equation.

Conclusion: The given quadratic equation has no real solutions due to the negative discriminant.

The correct answer to the problem is therefore, No solution.