Check whether the following are quadratic equations :
`(x - 2)^(2) +1 = 2x - 3`

Answer

To check whether the given equation \((x - 2)^{2} + 1 = 2x - 3\) is a quadratic equation, we will follow these steps:

### Step 1: Rewrite the equation
Start with the given equation:
\[ (x - 2)^{2} + 1 = 2x - 3 \]

### Step 2: Expand the left side
Expand \((x - 2)^{2}\):
\[ (x - 2)^{2} = x^{2} - 4x + 4 \]
So, substituting this back into the equation gives:
\[ x^{2} - 4x + 4 + 1 = 2x - 3 \]

### Step 3: Combine like terms
Combine the constants on the left side:
\[ x^{2} - 4x + 5 = 2x - 3 \]

### Step 4: Move all terms to one side
Rearranging the equation to bring all terms to one side results in:
\[ x^{2} - 4x - 2x + 5 + 3 = 0 \]
This simplifies to:
\[ x^{2} - 6x + 8 = 0 \]

### Step 5: Identify the standard form
The equation \(x^{2} - 6x + 8 = 0\) is now in the standard form of a quadratic equation, which is:
\[ ax^{2} + bx + c = 0 \]
where \(a = 1\), \(b = -6\), and \(c = 8\).

### Conclusion
Since the equation can be expressed in the standard form of a quadratic equation, we conclude that:
\[ \text{Yes, it is a quadratic equation.} \]

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