Check whether the following are quadratic equations :
x(2x + 3) = x + 2

Answer

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

### Step 1: Expand the left-hand side of the equation
We start with the equation:
\[ x(2x + 3) = x + 2 \]
Now, we will expand the left-hand side:
\[ 2x^2 + 3x = x + 2 \]

### Step 2: Rearrange the equation
Next, we will rearrange the equation to bring all terms to one side:
\[ 2x^2 + 3x - x - 2 = 0 \]
This simplifies to:
\[ 2x^2 + 2x - 2 = 0 \]

### Step 3: Factor out the common term
Now, we can factor out the common term from the left-hand side:
\[ 2(x^2 + x - 1) = 0 \]

### Step 4: Divide by the common factor
To simplify further, we can divide the entire equation by 2:
\[ x^2 + x - 1 = 0 \]

### Step 5: Identify the form of the equation
Now we have the equation in the standard form of a quadratic equation:
\[ ax^2 + bx + c = 0 \]
where \( a = 1 \), \( b = 1 \), and \( c = -1 \).

### Conclusion
Since the equation is in the form \( ax^2 + bx + c = 0 \) with \( a \neq 0 \), we conclude that:
\[ x(2x + 3) = x + 2 \text{ is a quadratic equation.} \]

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