https://file.aiquickdraw.com/mathbot/file/1726444708838ouf62n41.jpg

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!

Solve the equation: x^2/6 + (7 - x^2)/2 = -x^2/3

This problem involves solving a rational equation with quadratic terms. Learn how to combine fractions using a common denominator, simplify expressions, and recognize when an equation has no solution. Suitable for students in Grades 9-11.

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