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Integers are whole numbers that can be either positive, negative, or zero. These numbers do not have any fractional or decimal parts. For example, -3, 0, and 7 are integers, while 1.2 and 3/4 are not.
Integers are often represented on a number line, where zero is the central point. To the right of zero, numbers are positive, as they increase in value. To the left, numbers are negative, decreasing in value.
Integers have distinct properties that are handy in solving various algebraic equations, especially those involving simple mathematical operations like addition, subtraction, multiplication, and division (except by zero).

• Closure: The sum or product of any two integers is always an integer.
• Commutativity: For any two integers, the sum or product is the same regardless of their order.
• Associativity: The grouping of integers does not change their sum or product.
• Identity Elements: Zero is the additive identity because any integer added to zero remains unchanged, and one is the multiplicative identity because any integer multiplied by one remains unchanged.

Quadratic equations are mathematical equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(x\) represents the variable. The highest power of the variable \(x\) is 2, giving it the name "quadratic," derived from "quad," meaning square.
A key feature of quadratic equations is their graph, which is a parabola. The solutions to the quadratic equation, known as roots or zeros, are the points where the parabola intersects the x-axis.
In the exercise, the equation \(x^2 = x\) can be rewritten as \(x^2 - x = 0\), a form that is also quadratic, where:

• \(a = 1\)
• \(b = -1\)
• \(c = 0\)

The solutions occur when this equation equals zero. Here, factoring the equation gives \(x(x - 1) = 0\), leading to solutions for \(x = 0\) and \(x = 1\). These roots are the integers that solve the given problem.

Proof techniques are the methods used in mathematics to establish the truth of statements or theorems. They are essential to demonstrating claims logically and rigorously. In the exercise, the method used is the "existence proof," focusing on showing that at least one solution exists that satisfies the equation.
Existence proofs don't necessarily find all possible solutions or even provide a constructive method for finding them; they primarily care about proving that one or more solutions exist. This is done by providing specific examples of such solutions.

• Example: In our exercise, plugging values like \(x = 0\) and \(x = 1\) into the equation \(x^2 = x\), and confirming they work, proves their existence effectively.
• Applications: Such proofs are useful in theoretical computer science, pure mathematics, and other fields where showing the existence of solutions is often more important than finding all solutions.

By demonstrating existence, we show that the structure of the mathematical problem includes at least one option that meets the criteria. Thus, the existence proof reassures us of the validity of the statement under examination.