If x = 2 + √3/2 - √3 , determine the value of x^2 + x - 9 .

Question

If x = 2 + √3/2 - √3 , determine the value of x^2 + x - 9 .

1. A. 3√3
2. B. 5√3 (Correct answer)
3. C. 7√3
4. D. 9√3

Correct Answer

Option B — 5√3

Detailed Solution & Explanation

The correct answer is 5√3.

Key Points

• Rationalise inside the radical first: multiply top and bottom by $2 + \sqrt{3}$.
• $\dfrac{2 + \sqrt{3}}{2 - \sqrt{3}} = \dfrac{(2 + \sqrt{3})^2}{4 - 3} = (2 + \sqrt{3})^2$.
• Taking the square root gives $x = 2 + \sqrt{3}$, so $x^2 = 7 + 4\sqrt{3}$.
• Then $x^2 + x - 9 = (7 + 4\sqrt{3}) + (2 + \sqrt{3}) - 9 = 5\sqrt{3}$.

Additional Information

• The denominator $4 - 3 = 1$ is what makes the expression collapse so neatly — that is the designed shortcut.
• Numeric check: $x \approx 3.732$, $x^2 \approx 13.93$, and $13.93 + 3.73 - 9 \approx 8.66 = 5\sqrt{3}$ ✓
• Multiplying by the conjugate is the standard move whenever a surd sits in a denominator.

प्रश्न (हिन्दी में)

यदि x = 2 + √3/2 - √3 , तो x^2 + x - 9 का मान ज्ञात कीजिए।

1. A. 3√3
2. B. 5√3
3. C. 7√3
4. D. 9√3