91Ó°ÊÓ

Understanding how to solve equations is a foundational skill in mathematics. An equation represents a statement where two expressions are equal. Solving an equation means finding the value or set of values for the variable that makes the equation true.

Take, for instance, the equation from our exercise: \(x^2 = x\). To solve it, we need to isolate the variable \(x\). By subtracting \(x\) from both sides, we get \(x^2 - x = 0\). Here lies the intersection of solving equations with another crucial concept: factoring, which will be our next step in finding the solution. Finding the roots of the equation by setting each factor equal to zero enables us to arrive at the set of all possible solutions for \(x\), which in this case yields \(x = 0\) and \(x = 1\).

Factoring is a method of breaking down complex expressions into simpler ones that can easily be solved. It's an essential part of solving quadratic equations like the one in our exercise, \(x^2 - x = 0\). This equation can be factored into \(x(x - 1) = 0\), where the expression is written as a product of its factors.

The Zero Product Property states that if a product of factors equals zero, at least one of the factors must be zero. By applying this property, we set each factor equal to zero: \(x = 0\) and \(x - 1 = 0\), which gives us two possible solutions when solved individually. This leads to the conclusion that \(x\) can either be \(0\) or \(1\). Factoring is a powerful tool in equation solving that simplifies the process of finding the roots.

In set theory, comparing sets involves determining whether two sets have the same elements, a concept known as equality. Sets are considered equal if they contain precisely the same elements, no more and no less, regardless of the order in which these elements appear.

In our exercise, we compare the set of solutions from the equation \(x^2 = x\), which we've determined to be \(\{0, 1\}\), to the provided set \(\{0, 1\}\). To ascertain if these two sets are equal, we list their elements and observe if each element of one set is present in the other. As both sets contain only the numbers \(0\) and \(1\), and no others, we can conclude that these two sets are indeed equal. Comparing sets is a basic yet crucial operation in set theory, allowing for the classification of relationships between collections of objects.