Answer
414.9k+ views
Hint: We have a quadratic equation as: ${{x}^{2}}+px+q=0$ whose one root is $2+i\sqrt{3}$. So, we need to find the other root first. Then by using the sum of roots and product of roots formula, find the value of p and q.
Complete step by step answer:
As we know that, for a quadratic equation: $a{{x}^{2}}+bx+c=0$ , if one root is $\alpha +i\beta $ so the other root will be $\alpha -i\beta $
So, for the quadratic equation given in the question: ${{x}^{2}}+px+q=0$. Since one root is $2+i\sqrt{3}$. Therefore, the other root will be $2-i\sqrt{3}$
Now, we know that: for a quadratic equation: $a{{x}^{2}}+bx+c=0$
Sum of roots is: $-\dfrac{b}{a}$
Product of roots is: \[\dfrac{c}{a}\]
So, for the quadratic equation given in the question: ${{x}^{2}}+px+q=0$
Sum of roots is: $-\dfrac{p}{1}......(1)$
Product of roots is: \[\dfrac{q}{1}......(1)\]
Also, we know that the roots of the quadratic equation are: $2+i\sqrt{3}$ and $2-i\sqrt{3}$
So, we can write equation (1) and equation (2) as:
\[\begin{align}
& 2+i\sqrt{3}+2-i\sqrt{3}=-p......(3) \\
& \left( 2+i\sqrt{3} \right)\left( 2-i\sqrt{3} \right)=q......(4) \\
\end{align}\]
By solving equation (4) and equation (5) using $i=\sqrt{-1}$ , we get the value of p and q as:
$\begin{align}
& \Rightarrow 2+2=-p \\
& p=-4 \\
& \Rightarrow \left( 2\times 2 \right)+\left( 2\times \left( -i\sqrt{3} \right) \right)+\left( i\sqrt{3}\times 2 \right)+\left( i\sqrt{3}\times \left( -i\sqrt{3} \right) \right)=q \\
& 4-2i\sqrt{3}+2i\sqrt{3}+3=q \\
& q=7 \\
\end{align}$
So, $\left( p,q \right)=\left( -4,7 \right)$
So, the correct answer is “Option A”.
Note: While applying the identity for the sum of zeros and products of zeros, always take care of the negative sign in the sum of zeros. Some deliberately miss out on the use of negative signs in the formula and this gives you the wrong value. Also, it was given in the question, that the polynomial is quadratic. For a higher degree of the polynomial, the formula for the sum of zeros and product of zeros changes accordingly.
Complete step by step answer:
As we know that, for a quadratic equation: $a{{x}^{2}}+bx+c=0$ , if one root is $\alpha +i\beta $ so the other root will be $\alpha -i\beta $
So, for the quadratic equation given in the question: ${{x}^{2}}+px+q=0$. Since one root is $2+i\sqrt{3}$. Therefore, the other root will be $2-i\sqrt{3}$
Now, we know that: for a quadratic equation: $a{{x}^{2}}+bx+c=0$
Sum of roots is: $-\dfrac{b}{a}$
Product of roots is: \[\dfrac{c}{a}\]
So, for the quadratic equation given in the question: ${{x}^{2}}+px+q=0$
Sum of roots is: $-\dfrac{p}{1}......(1)$
Product of roots is: \[\dfrac{q}{1}......(1)\]
Also, we know that the roots of the quadratic equation are: $2+i\sqrt{3}$ and $2-i\sqrt{3}$
So, we can write equation (1) and equation (2) as:
\[\begin{align}
& 2+i\sqrt{3}+2-i\sqrt{3}=-p......(3) \\
& \left( 2+i\sqrt{3} \right)\left( 2-i\sqrt{3} \right)=q......(4) \\
\end{align}\]
By solving equation (4) and equation (5) using $i=\sqrt{-1}$ , we get the value of p and q as:
$\begin{align}
& \Rightarrow 2+2=-p \\
& p=-4 \\
& \Rightarrow \left( 2\times 2 \right)+\left( 2\times \left( -i\sqrt{3} \right) \right)+\left( i\sqrt{3}\times 2 \right)+\left( i\sqrt{3}\times \left( -i\sqrt{3} \right) \right)=q \\
& 4-2i\sqrt{3}+2i\sqrt{3}+3=q \\
& q=7 \\
\end{align}$
So, $\left( p,q \right)=\left( -4,7 \right)$
So, the correct answer is “Option A”.
Note: While applying the identity for the sum of zeros and products of zeros, always take care of the negative sign in the sum of zeros. Some deliberately miss out on the use of negative signs in the formula and this gives you the wrong value. Also, it was given in the question, that the polynomial is quadratic. For a higher degree of the polynomial, the formula for the sum of zeros and product of zeros changes accordingly.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)