Question

If \(x = \sqrt{3}\) is a solution of the equation \(ax^2 + \sqrt{3}x - 12 = 0\), then :

• A. a = 3
• B. a = 2
• C. a = 1
• D. a = 3

Hint:

Whenever a root is given in irrational form (such as \(\sqrt{3}\)), be extremely careful when performing squaring operations and products.
Squaring a square root yields the base integer directly, which simplifies the equation quickly into a basic linear equation in terms of the unknown variable.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given a quadratic equation \(ax^2 + \sqrt{3}x - 12 = 0\) and told that \(x = \sqrt{3}\) is one of its roots (solutions). We need to determine the value of the coefficient \(a\).

Step 2: Key Formula or Approach:
By definition, if a value \(x = \alpha\) is a solution to the equation \(f(x) = 0\), then substituting \(\alpha\) in place of \(x\) must satisfy the equation. That is, \(f(\alpha) = 0\).

Step 3: Detailed Explanation:
1. Substitute \(x = \sqrt{3}\) into the given equation \(ax^2 + \sqrt{3}x - 12 = 0\):
\[ a(\sqrt{3})^2 + \sqrt{3}(\sqrt{3}) - 12 = 0 \]
2. Simplify individual terms of the equation:
We know that \((\sqrt{3})^2 = 3\).
We also know that \(\sqrt{3} \times \sqrt{3} = 3\).
3. Substituting these simplified values back into the equation yields:
\[ a(3) + 3 - 12 = 0 \]
\[ 3a - 9 = 0 \]
4. Solve the linear equation for \(a\):
\[ 3a = 9 \]
\[ a = \frac{9}{3} = 3 \]
This gives the value of the unknown coefficient \(a\) as 3.

Step 4: Final Answer:
The value of \(a\) is 3, which corresponds to option (A).