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

Options

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

Answer

To factorise the expression \(4x^2 - 12x + 9\), we will 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: Multiply \(a\) and \(c\)
We need to multiply \(a\) and \(c\):
\[ a \cdot c = 4 \cdot 9 = 36 \]

### Step 3: Find two numbers that multiply to \(ac\) and add to \(b\)
We need to find two numbers that multiply to \(36\) (the result from Step 2) and add to \(-12\) (the value of \(b\)). The numbers that satisfy these conditions are \(-6\) and \(-6\) because:
\[ -6 \cdot -6 = 36 \quad \text{and} \quad -6 + -6 = -12 \]

### Step 4: Rewrite the middle term
We can rewrite the expression by splitting the middle term using the two numbers found:
\[ 4x^2 - 6x - 6x + 9 \]

### Step 5: Factor by grouping
Now we will group the terms:
\[ (4x^2 - 6x) + (-6x + 9) \]
Now, factor out the common factors from each group:
\[ 2x(2x - 3) - 3(2x - 3) \]

### Step 6: Factor out the common binomial
Now we can factor out the common binomial \((2x - 3)\):
\[ (2x - 3)(2x - 3) \]
This can also be written as:
\[ (2x - 3)^2 \]

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

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