Match the equation in Column I with its solution(s) in Column II. x^2 = -25

Verified step by step guidance

1. Recognize that the equation \(x^2 = -25\) involves a negative number on the right side.
2. Recall that the square of a real number is always non-negative, so \(x^2 = -25\) has no real solutions.
3. Consider the possibility of complex solutions, since the equation involves a negative number.
4. Rewrite the equation as \(x^2 = -25\) and take the square root of both sides, remembering to include the imaginary unit \(i\) for the negative square root.
5. The solutions are \(x = 5i\) and \(x = -5i\), where \(i\) is the imaginary unit defined as \(i^2 = -1\).

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where 'i' is the imaginary unit defined as the square root of -1. They are essential for solving equations that do not have real solutions, such as x^2 = -25, which leads to solutions involving imaginary numbers.

Square Roots of Negative Numbers

When taking the square root of a negative number, the result is not a real number but an imaginary number. For example, the equation x^2 = -25 implies that x = ±√(-25), which can be simplified to x = ±5i, where 'i' represents the imaginary unit. Understanding this concept is crucial for solving equations with negative results.

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants. The solutions to quadratic equations can be found using various methods, including factoring, completing the square, or applying the quadratic formula. In this case, recognizing that the equation x^2 = -25 is a quadratic equation helps in identifying the appropriate methods for finding its solutions.