In algebra, a real solution refers to a value that satisfies an equation and is a real number, meaning it can be found on the number line. To understand this, we need to distinguish between different types of numbers: real numbers include all the rational and irrational numbers, while complex numbers involve the imaginary unit ‘i’, which is used to represent the square root of negative one.
When we solve an algebraic equation, we are often looking for the points where the equation intersects the x-axis (for a function) or where the equation holds true. If we find a solution that is a real number, we can graph it visually, displaying it as a point on the number line. For example, if we have the equation x² – 4 = 0, we can factor it into (x – 2)(x + 2) = 0, yielding the real solutions x = 2 and x = -2.
In contrast, if an equation yields complex solutions, such as x² + 1 = 0 (which has solutions of x = i and x = -i), those do not correspond to any points on the standard real number line and are not considered real solutions.
In summary, a real solution is a value that solves an algebraic equation and is part of the set of real numbers, important for understanding the behavior of functions and equations in various mathematical contexts.