Solve the Quadratic Equation with Fractions: x²/2 - x + 2/3 = 0

Solve the following equation:

$\frac{x^2}{2}-x+\frac{2}{3}=0$

To solve this problem, we'll follow these steps:

• Step 1: Identify the provided equation and standardize it.
• Step 2: Determine the coefficients $a$, $b$, and $c$.
• Step 3: Compute the discriminant $b^2 - 4ac$.
• Step 4: Analyze the discriminant to determine the nature of the solutions.

Let's work through each step:

Step 1: The given equation is $\frac{x^2}{2} - x + \frac{2}{3} = 0$. For simplicity, we multiply through by 6 to clear fractions:
This becomes $3x^2 - 6x + 4 = 0$.

Step 2: Identify coefficients for the quadratic formula:
$a = 3$, $b = -6$, and $c = 4$.

Step 3: Compute the discriminant $b^2 - 4ac$:
Discriminant $= (-6)^2 - 4 \times 3 \times 4 = 36 - 48 = -12$.

Step 4: Analyze the discriminant:
The discriminant is negative ($-12$), indicating no real solutions.

No solution exists for this equation in the real number set.

Therefore, the solution to the problem is: No solution.