The expressions `4x^(2)-12` is equivalent too `4(x+c)(x-c)`. If c is a positive constant, what is the value of c?

Options

A. `sqrt(3)`
B. `3`
C. `sqrt(12)`
D. `12`

Answer

To solve the problem, we need to find the value of \( c \) such that the expression \( 4x^2 - 12 \) is equivalent to \( 4(x+c)(x-c) \).

### Step-by-Step Solution:

1. **Start with the given expression:**
\[ 4x^2 - 12 \]

2. **Factor out the common term from the left side:**
\[ 4(x^2 - 3) \]
Here, we notice that \( 12 = 4 \times 3 \), so we can rewrite \( -12 \) as \( -4 \times 3 \).

3. **Rewrite the right side using the difference of squares:**
The expression \( 4(x+c)(x-c) \) can be expanded using the formula \( a^2 - b^2 = (a+b)(a-b) \):
\[ 4(x^2 - c^2) \]

4. **Set the two expressions equal to each other:**
\[ 4(x^2 - 3) = 4(x^2 - c^2) \]

5. **Cancel the common factor of 4 from both sides:**
\[ x^2 - 3 = x^2 - c^2 \]

6. **Eliminate \( x^2 \) from both sides:**
\[ -3 = -c^2 \]

7. **Multiply through by -1:**
\[ 3 = c^2 \]

8. **Take the square root of both sides:**
\[ c = \sqrt{3} \]
Since \( c \) is specified to be a positive constant, we take the positive root.

### Final Answer:
\[ c = \sqrt{3} \]