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Real solutions are the values of a variable that satisfy an equation to make it true under the real number system. For quadratic equations, which are generally written in the form \(ax^2 + bx + c = 0\), the solutions can be determined using a variety of methods, such as factoring, using the quadratic formula, or completing the square.

Quadratic equations can have:

• Two distinct real solutions: This occurs when the graph of the equation (a parabola) intersects the x-axis at two different points.
• One real solution: This is also known as a repeated or double root, occurring when the parabola just touches the x-axis.
• No real solutions: This happens when the parabola does not intersect the x-axis at all, indicating that the solutions are complex numbers and not real.

In our specific case, the original equation was simplified to \(x^2 = 4\), where the right side of the equation is a positive number. This means that there are real solutions because the square of a real number can be both positive or zero.

Solving quadratic equations involves finding the values of the unknown variable that make the equation true. It often starts by simplifying the equation, if needed, to the form \(ax^2 + bx + c = 0\). Then, various methods and strategies can be applied:

• Factoring: This involves expressing the quadratic as a product of two binomials. For example, the equation \(x^2 - 5x + 6 = 0\) can be factored into \((x - 2)(x - 3) = 0\).
• Quadratic formula: This is generally used when factoring is not straightforward. The formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) provides solutions directly.
• Completing the square: This method rewrites the equation so that one side is a perfect square trinomial. From there, you can solve for the variable by taking the square root of both sides.

For the example given in the original exercise, the equation \(x^2 = 4\) is already simplified, making it clear that the solutions are in the set of real numbers that, when squared, equal 4. These solutions are \(x = 2\) and \(x = -2\).

Non-negative numbers include all the numbers that are greater than or equal to zero—this means zero and all positive numbers. Understanding non-negative numbers is key in determining the existence of solutions for quadratic equations. In the context of the equation \(x^2 = a\), it is important that \(a\) is non-negative for the solution to exist in the real number system.

This is because the square of a real number cannot be negative. Hence, a quadratic equation like \(x^2 = -4\) would not have any real solutions since no real number squared yields a negative result.

In the original exercise, checking whether \(a\), which was 4 in the simplified equation \(x^2 = 4\), was non-negative confirmed the availability of real solutions. Hence, whenever you solve a quadratic equation, always ensure the constant on one side is non-negative to ascertain the potential for real solutions.