Is `(3x-2)(2x-3)=(2x+5)(2x-1)` a quadratic equation ?

Answer

To determine whether the equation \((3x-2)(2x-3) = (2x+5)(2x-1)\) is a quadratic equation, we will follow these steps:

### Step 1: Expand both sides of the equation

First, we will expand the left-hand side:

\[ (3x - 2)(2x - 3) = 3x \cdot 2x + 3x \cdot (-3) - 2 \cdot 2x - 2 \cdot (-3) \]

Calculating this gives:

\[ = 6x^2 - 9x - 4x + 6 \]
\[ = 6x^2 - 13x + 6 \]

Now, we will expand the right-hand side:

\[ (2x + 5)(2x - 1) = 2x \cdot 2x + 2x \cdot (-1) + 5 \cdot 2x + 5 \cdot (-1) \]

Calculating this gives:

\[ = 4x^2 - 2x + 10x - 5 \]
\[ = 4x^2 + 8x - 5 \]

### Step 2: Set the equation to zero

Now we set both sides equal to each other:

\[ 6x^2 - 13x + 6 = 4x^2 + 8x - 5 \]

Next, we will move all terms to one side of the equation:

\[ 6x^2 - 13x + 6 - 4x^2 - 8x + 5 = 0 \]

Combining like terms:

\[ (6x^2 - 4x^2) + (-13x - 8x) + (6 + 5) = 0 \]
\[ 2x^2 - 21x + 11 = 0 \]

### Step 3: Identify the form of the equation

The resulting equation is:

\[ 2x^2 - 21x + 11 = 0 \]

This is in the standard form of a quadratic equation, which is \(ax^2 + bx + c = 0\), where \(a = 2\), \(b = -21\), and \(c = 11\).

### Conclusion

Since \(a \neq 0\), we conclude that the equation is indeed a quadratic equation.