Which of the following is equivalent to
[tex]\frac{4x^2 - 3}{2x + \sqrt{3}}?[/tex]

A. [tex]2x - \sqrt{3}[/tex]
B. [tex]2x + \sqrt{3}[/tex]
C. [tex]2x - 3[/tex]
D. [tex]2x + 3[/tex]



Answer :

To determine which option is equivalent to [tex]\(\frac{4x^2 - 3}{2x + \sqrt{3}}\)[/tex], we need to verify each option by simplifying the expression.

First, let's rewrite the original expression for clarity:
[tex]\[ \frac{4x^2 - 3}{2x + \sqrt{3}} \][/tex]

We need to find which of the given options equals this expression when simplified.

### Option A: [tex]\(2x - \sqrt{3}\)[/tex]
Let's check if [tex]\(2x - \sqrt{3}\)[/tex] is equivalent to [tex]\(\frac{4x^2 - 3}{2x + \sqrt{3}}\)[/tex].

1. Multiply both sides of the equation by [tex]\(2x + \sqrt{3}\)[/tex] to clear the denominator:
[tex]\[ 4x^2 - 3 = (2x - \sqrt{3})(2x + \sqrt{3}) \][/tex]

2. Simplify the right-hand side:
[tex]\[ (2x - \sqrt{3})(2x + \sqrt{3}) = 2x \cdot 2x + 2x \cdot \sqrt{3} - \sqrt{3} \cdot 2x - \sqrt{3} \cdot \sqrt{3} \][/tex]
[tex]\[ = 4x^2 + 2x\sqrt{3} - 2x\sqrt{3} - 3 \][/tex]
[tex]\[ = 4x^2 - 3 \][/tex]

The resulting expression is:
[tex]\[ 4x^2 - 3 \][/tex]

This matches the left-hand side of our original equation, confirming that:
[tex]\[ 2x - \sqrt{3} \][/tex]
is indeed equivalent to [tex]\(\frac{4x^2 - 3}{2x + \sqrt{3}}\)[/tex].

### Verification of Other Options

For completeness, let's briefly check the other options:

- Option B: [tex]\(2x + \sqrt{3}\)[/tex]:
[tex]\[ \frac{4x^2 - 3}{2x + \sqrt{3}} \cdot (2x + \sqrt{3}) \neq 4x^2 - 3 \quad \text{unless simplified operations match LHS, which they don't in correct meaning.} \][/tex]
This yields [tex]\(4x^2 + 3\)[/tex], which is incorrect.

- Option C: [tex]\(2x - 3\)[/tex]:
[tex]\[ \frac{4x^2 - 3}{2x + \sqrt{3}} \cdot (2x + \sqrt{3}) \neq 4x^2 - 3 \quad \text{unless simplified operations match LHS, which they don't in correct meaning.} \][/tex]

- Option D: [tex]\(2x + 3\)[/tex]:
[tex]\[ \frac{4x^2 - 3}{2x + \sqrt{3}} \cdot (2x + \sqrt{3}) \neq 4x^2 - 3 \quad \text{unless simplified operations match LHS, which they don't in correct meaning.} \][/tex]

Thus, from the consideration of the steps and verifying equivalence:
[tex]\[ \boxed{A: 2x - \sqrt{3}} \][/tex]

is the correct answer.

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