Question

List all the subsets of the set: x where x is the solution of ${x^2} + 5x + 6 = 0$.

Answer 1

A set is a collection of objects where each object in the set is called an element for that set denoted by $x \in R$ where x is the element of the set R and set having no elements to them is called an empty set.
An interval is a set number that consists of all real numbers between a given pair of numbers. An endpoint of an interval is either of the two points that mark the endpoint of a line segment. An interval can be of different types which can include either endpoint or both endpoints or neither endpoint.
In this question, we have to list down all the sub-set of x, which is the solution of the quadratic equation ${x^2} + 5x + 6 = 0$. For this, we will first carry out the method of splitting the middle term and find the roots of the equation.

Complete step by step answer:
The given quadratic equation is ${x^2} + 5x + 6 = 0$
Following the method of splitting the middle term in the quadratic equation ${x^2} + 5x + 6 = 0$, we get:
{x^2} + 5x + 6 = 0 \\\ {x^2} + 3x + 2x + 6 = 0 - - - - (i) \\\
Take the common terms out in the equation (i) as:
{x^2} + 3x + 2x + 6 = 0 \\\ x(x + 3) + 2(x + 3) = 0 \\\ (x + 2)(x + 3) = 0 - - - - (ii) \\\
Now, equating each of the two terms to zero of the equation (iii) to get the solution of the quadratic equation as:
x + 2 = 0;x + 3 = 0 \\\ x = - 2;x = - 3 \\\
Hence, the solution to the quadratic equation ${x^2} + 5x + 6 = 0$ is $x = - 2; - 3$.

So, the sub-set of the set x is $\\{ - 2, - 3\\}$.

Note: Students often get confused with the terms sub-set as the total values that are included in both the sets, which is not true. Sub-set contains only the common terms which are present in the given sets.