Factorise `2x ^(2) + sqrt3 x - 3.`

Options

A. `(x + sqrt3) ( 2x + sqrt3)`
B. `(x - sqrt3) ( 2x - sqrt3)`
C. `(x + sqrt3) ( 2x - sqrt3)`
D. `(x - sqrt3) (x - sqrt3)`

Answer

To factorize the expression \(2x^2 + \sqrt{3}x - 3\), we can follow these steps:

### Step 1: Identify the coefficients
The given quadratic expression is in the form \(ax^2 + bx + c\), where:
- \(a = 2\)
- \(b = \sqrt{3}\)
- \(c = -3\)

### Step 2: Multiply \(a\) and \(c\)
Calculate the product of \(a\) and \(c\):
\[ a \cdot c = 2 \cdot (-3) = -6 \]

### Step 3: Find two numbers that multiply to \(ac\) and add to \(b\)
We need to find two numbers that multiply to \(-6\) (the result from step 2) and add up to \(\sqrt{3}\) (the value of \(b\)).

After checking possible pairs, we find:
- The numbers \(2\sqrt{3}\) and \(-\sqrt{3}\) satisfy:
\[ 2\sqrt{3} \cdot (-\sqrt{3}) = -6 \quad \text{and} \quad 2\sqrt{3} + (-\sqrt{3}) = \sqrt{3} \]

### Step 4: Rewrite the middle term
Now, we can rewrite the expression by splitting the middle term using the two numbers found:
\[ 2x^2 + 2\sqrt{3}x - \sqrt{3}x - 3 \]

### Step 5: Group the terms
Group the terms into two pairs:
\[ (2x^2 + 2\sqrt{3}x) + (-\sqrt{3}x - 3) \]

### Step 6: Factor out the common factors in each group
From the first group \(2x^2 + 2\sqrt{3}x\), we can factor out \(2x\):
\[ 2x(x + \sqrt{3}) \]
From the second group \(-\sqrt{3}x - 3\), we can factor out \(-\sqrt{3}\):
\[ -\sqrt{3}(x + \sqrt{3}) \]

### Step 7: Combine the factored terms
Now we can combine the factored terms:
\[ 2x(x + \sqrt{3}) - \sqrt{3}(x + \sqrt{3}) = (x + \sqrt{3})(2x - \sqrt{3}) \]

### Final Result
Thus, the factorization of the expression \(2x^2 + \sqrt{3}x - 3\) is:
\[ (x + \sqrt{3})(2x - \sqrt{3}) \]