text: # Solve this following
  
  Question:
  
  Let $p, q \in R$. If $2-\sqrt{3}$ is a root of the quadratic
  
  equation, $x^{2}+p x+q=0$, then :
  
  1. $\mathrm{q}^{2}+4 \mathrm{p}+14=0$
  
  2. $\mathrm{p}^{2}-4 \mathrm{q}-12=0$
  
  3. $q^{2}-4 p-16=0$
  
  4. $p^{2}-4 q+12=0$
  
  Correct Option: , 2
  
  Solution:
  
  In given question $\mathrm{p}, \mathrm{q} \in \mathrm{R}$. If we take other root as any real number $\alpha$, then quadratic equation will be
  
  $x^{2}-(\alpha+2-\sqrt{3}) x+\alpha \cdot(2-\sqrt{3})=0$
  
  Now, we can have none or any of the options can be correct depending upon ' $\alpha$ ' Instead of $\mathrm{p}, \mathrm{q} \in \mathrm{R}$ it should be $\mathrm{p}, \mathrm{q} \in \mathrm{Q}$ then other root will be $2+\sqrt{3}$
  
  $\Rightarrow \mathrm{p}=-(2+\sqrt{3}-2-\sqrt{3})=-4$
  
  and $\mathrm{q}=(2+\sqrt{3})(2-\sqrt{3})=1$
  
  $\Rightarrow \mathrm{p}^{2}-4 \mathrm{q}-12=(-4)^{2}-4-12$
  
  $=16-16=0$
  
  Option (2) is correct
explanation: The document contains a math question and its corresponding solution. The question is about determining the correct relationship between the coefficients \( p \) and \( q \) of a quadratic equation given that \( 2 - \sqrt{3} \) is a root. The solution provides a detailed step-by-step reasoning to arrive at the correct option.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Solve this following
  
  Question:
  
  Let $p, q \in R$. If $2-\sqrt{3}$ is a root of the quadratic
  
  equation, $x^{2}+p x+q=0$, then :
  
  1. $\mathrm{q}^{2}+4 \mathrm{p}+14=0$
  
  2. $\mathrm{p}^{2}-4 \mathrm{q}-12=0$
  
  3. $q^{2}-4 p-16=0$
  
  4. $p^{2}-4 q+12=0$
  
  Correct Option: , 2
  
  Solution:
  
  In given question $\mathrm{p}, \mathrm{q} \in \mathrm{R}$. If we take other root as any real number $\alpha$, then quadratic equation will be
  
  $x^{2}-(\alpha+2-\sqrt{3}) x+\alpha \cdot(2-\sqrt{3})=0$
  
  Now, we can have none or any of the options can be correct depending upon ' $\alpha$ ' Instead of $\mathrm{p}, \mathrm{q} \in \mathrm{R}$ it should be $\mathrm{p}, \mathrm{q} \in \mathrm{Q}$ then other root will be $2+\sqrt{3}$
  
  $\Rightarrow \mathrm{p}=-(2+\sqrt{3}-2-\sqrt{3})=-4$
  
  and $\mathrm{q}=(2+\sqrt{3})(2-\sqrt{3})=1$
  
  $\Rightarrow \mathrm{p}^{2}-4 \mathrm{q}-12=(-4)^{2}-4-12$
  
  $=16-16=0$
  
  Option (2) is correct
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and its corresponding solution. The question is about determining the correct relationship between the coefficients \( p \) and \( q \) of a quadratic equation given that \( 2 - \sqrt{3} \) is a root. The solution provides a detailed step-by-step reasoning to arrive at the correct option.
  
  Label: question-and-answer