Question

Find the equation whose sum of roots and product of roots are the product and sum of roots of \[{x^2} + 5x + 6 = 0\] respectively.
A) \[{x^2} - 6x - 5 = 0\]
B) \[{x^2} - 5x - 6 = 0\]
C) \[{x^2} + 11x - 1 = 0\]
D) None of the above

Answer 1

Hint:
Here we will first find the sum and product of roots of the given equation; from there we will get the sum and product of the required equation. Then to find the equation, we will use the standard form of quadratic equation. We will substitute the value of sum of roots and product of roots in the standard equation to get the required quadratic equation.

Complete step by step solution:
The given quadratic equation is \[{x^2} + 5x + 6 = 0\].
Now, we will find the sum of roots of the given quadratic equation.
We know, sum of roots of the quadratic equation is equal to the negative of the ratio of the coefficient of \[x\] to the coefficient of\[{x^2}\].
Therefore,
\[ \Rightarrow {\rm{sum}} = - \dfrac{5}{1} = - 5\]
Now, we will find the product of roots of the given quadratic equation.
We know, product of roots of a quadratic equation is equal to the ratio of the constant term to the coefficient of \[{x^2}\].
Therefore,
\[ \Rightarrow {\rm{product}} = \dfrac{6}{1} = 6\]
It is given that the sum of roots of the required equation is equal to the product of roots of the given equation. Also, the product of roots of the required equation is equal to the sum of roots of the given equation.
Therefore,
Sum of roots of the required equation is equal to 6 and the product of roots of the required equation is equal to \[ - 5\].
We know the standard form of the quadratic equation which is \[{x^2} - Sx + P = 0\]; where \[S\] is the sum of roots and \[P\] is the product of roots.
Substituting the value of sum of roots and product of roots in the standard equation, we get
\[\begin{array}{l} \Rightarrow {x^2} - \left( 6 \right)x + \left( { - 5} \right) = 0\\ \Rightarrow {x^2} - 6x - 5 = 0\end{array}\]

Hence, the correct option is option A.

Note:
An equation is said to be quadratic if the highest value of the variable is two. Here we have obtained the equation with the help of the roots of the equation. Roots of the quadratic equation are defined as the values of \[x\] which when substituted in the equation satisfies the equation. A quadratic equation has only 2 roots, a cubic equation has 3 roots and so on.