Factorise `4x^(2) -12x + 9`

Options

A. `(2x -3)^(2)`
B. `(2x+3)^(2)`
C. `(3x+2)^(2)`
D. `(3x-2)^(2)`

Answer

To factorize the expression \(4x^2 - 12x + 9\), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic expression is in the form \(ax^2 + bx + c\), where:
- \(a = 4\)
- \(b = -12\)
- \(c = 9\)

### Step 2: Rewrite the quadratic
We can express \(4x^2\) as \((2x)^2\) since \(4\) is the square of \(2\). Thus, we rewrite the expression:
\[ 4x^2 - 12x + 9 = (2x)^2 - 12x + 9 \]

### Step 3: Recognize the perfect square trinomial
Next, we notice that \(9\) can be expressed as \(3^2\). The expression now looks like:
\[ (2x)^2 - 2 \cdot (2x) \cdot 3 + 3^2 \]
This is in the form of a perfect square trinomial, which is \(A^2 - 2AB + B^2 = (A - B)^2\).

### Step 4: Apply the perfect square formula
Here, let \(A = 2x\) and \(B = 3\). Thus, we can factor the expression as:
\[ (2x - 3)^2 \]

### Final Answer
Therefore, the factorization of \(4x^2 - 12x + 9\) is:
\[ (2x - 3)^2 \]

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