Final answer:
Without a specific expression or equation, general statements can be made about finding rational or irrational values for x. For quadratic equations, the quadratic formula can give rational solutions if the discriminant is a perfect square, and irrational solutions otherwise. Specific examples include x = ±√2 as irrational solutions and x = ± 2 as rational solutions.
Step-by-step explanation:
To find values for x that make the expression either rational or irrational, we need to first have a clear expression or equation to evaluate. Unfortunately, the provided question does not give a specific expression or equation to work with. However, generally speaking, rational numbers are those that can be expressed as the ratio of two integers (e.g., fractions), while irrational numbers cannot be expressed as such ratios and have non-repeating, non-terminating decimal expansions.
For a quadratic equation of the form x² + bx + c = 0, the quadratic formula, x = (-b ± √(b²-4ac)) / (2a), can be used to find the solutions. Substituting the given coefficients into the quadratic formula would yield two values for x. If both the discriminant (b²-4ac) and the solutions are rational numbers, then we have found two rational values for x.
Alternatively, if the discriminant is not a perfect square, the solutions for x will typically be irrational, since they involve taking the square root of an irrational number. If you provide a specific expression or equation, more accurate values for x can be given, which can be rational or irrational depending on the exact nature of the equation.
For example, if we have the equation x² - 2 = 0, the solutions would be x = ±√2, which are irrational. However, if we had x² - 4 = 0, the solutions would be x = ± 2, which are rational.
Rational and irrational numbers can also arise in complex expressions involving square roots, exponents, and other operations. When checking your answers, make sure they are reasonable and correspond with the original problem constraints.