The equation provided in the image is:

$\frac{x^2}{6} + \frac{7 - x^2}{2} = -\frac{x^2}{3}$

Let's solve this step by step:

Step 1: Simplify the equation

First, rewrite the terms so that they all have a common denominator.

$\frac{x^2}{6} + \frac{7 - x^2}{2} = -\frac{x^2}{3}$

The least common denominator (LCD) of 6, 2, and 3 is 6. Rewrite all terms with a denominator of 6:

$\frac{x^2}{6} + \frac{3(7 - x^2)}{6} = \frac{-2x^2}{6}$

This simplifies to:

$\frac{x^2}{6} + \frac{21 - 3x^2}{6} = \frac{-2x^2}{6}$

Step 2: Combine terms on the left-hand side

Now that all terms have the same denominator, combine them:

$\frac{x^2 + (21 - 3x^2)}{6} = \frac{-2x^2}{6}$

Simplify the numerator:

$\frac{x^2 + 21 - 3x^2}{6} = \frac{-2x^2}{6}$

$\frac{-2x^2 + 21}{6} = \frac{-2x^2}{6}$

Step 3: Eliminate the denominator

Multiply both sides of the equation by 6 to eliminate the denominator:

$-2x^2 + 21 = -2x^2$

Step 4: Solve for $x^2$

Cancel the $-2x^2$ on both sides:

$21 = 0$

This leads to a contradiction, which means there is no solution to the equation.

Would you like more details, or do you have any other questions? Here are some related questions you could explore:

1. How do you find the least common denominator in a fraction equation?
2. What does it mean if an equation has no solution?
3. How can contradictions arise when solving quadratic equations?
4. Could this equation have a solution in a different context, like complex numbers?
5. What are the common types of errors that lead to contradictions when solving equations?

Tip: Always check for contradictions in the final steps of algebraic simplification. It could indicate that no solution exists for the given equation!