`4z ^(2)- 8z + 3= ? `

Options

A. `(2z - 1) (2z -3)`
B. `(2z + 10 (3-2z)`
C. `(2z + 3) (3z -1)`
D. `(z -1) (4z-3)`

Answer

To factor the expression \(4z^2 - 8z + 3\), we can follow these steps:

### Step 1: Identify the coefficients
The given expression is \(4z^2 - 8z + 3\). Here, the coefficients are:
- Coefficient of \(z^2\) (a) = 4
- Coefficient of \(z\) (b) = -8
- Constant term (c) = 3

### Step 2: Multiply \(a\) and \(c\)
Next, we multiply the coefficient of \(z^2\) (which is 4) by the constant term (which is 3):
\[ a \cdot c = 4 \cdot 3 = 12 \]

### Step 3: Find two numbers that multiply to \(ac\) and add to \(b\)
We need to find two numbers that multiply to 12 and add up to -8. The two numbers that satisfy this are -6 and -2:
\[ -6 \cdot -2 = 12 \quad \text{and} \quad -6 + (-2) = -8 \]

### Step 4: Rewrite the middle term
Now we can rewrite the expression by breaking down the middle term (-8z) using the two numbers we found:
\[ 4z^2 - 6z - 2z + 3 \]

### Step 5: Group the terms
Next, we group the terms in pairs:
\[ (4z^2 - 6z) + (-2z + 3) \]

### Step 6: Factor out the common terms
Now we factor out the common factors from each group:
1. From the first group \(4z^2 - 6z\), we can factor out \(2z\):
\[ 2z(2z - 3) \]
2. From the second group \(-2z + 3\), we can factor out \(-1\):
\[ -1(2z - 3) \]

Putting it all together, we have:
\[ 2z(2z - 3) - 1(2z - 3) \]

### Step 7: Factor out the common binomial
Now we can factor out the common binomial \((2z - 3)\):
\[ (2z - 3)(2z - 1) \]

### Final Answer
Thus, the factored form of \(4z^2 - 8z + 3\) is:
\[ (2z - 3)(2z - 1) \]

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