is `3x(2x-5)+6=2x(3x+5)-6` a quadratic equation ?

Answer

To determine if the equation \(3x(2x-5)+6=2x(3x+5)-6\) is a quadratic equation, we will simplify both sides and check the degree of the resulting equation.

### Step-by-step Solution:

1. **Write the given equation**:
\[ 3x(2x - 5) + 6 = 2x(3x + 5) - 6 \]

2. **Expand both sides**:
- Left side:
\[ 3x(2x - 5) = 6x^2 - 15x \]
Therefore, the left side becomes:
\[ 6x^2 - 15x + 6 \]
- Right side:
\[ 2x(3x + 5) = 6x^2 + 10x \]
Therefore, the right side becomes:
\[ 6x^2 + 10x - 6 \]

3. **Set the equation with both sides expanded**:
\[ 6x^2 - 15x + 6 = 6x^2 + 10x - 6 \]

4. **Move all terms to one side**:
Subtract \(6x^2\) from both sides:
\[ -15x + 6 = 10x - 6 \]
Now, move \(10x\) and \(-6\) to the left side:
\[ -15x - 10x + 6 + 6 = 0 \]
This simplifies to:
\[ -25x + 12 = 0 \]

5. **Rearranging the equation**:
\[ -25x + 12 = 0 \]
This can be rewritten as:
\[ 25x = 12 \]

6. **Check the degree of the equation**:
The equation \(25x = 12\) is a linear equation (degree 1) because the highest power of \(x\) is 1.

### Conclusion:
Since the highest degree of \(x\) in the equation is 1, we conclude that the given equation is **not a quadratic equation**.