Match the equation in Column I with its solution(s) in Column II. x² = -25

Verified step by step guidance

1. Recognize that the equation given is \(x^2 = -25\), which is a quadratic equation where the square of \(x\) equals a negative number.
2. Recall that in the set of real numbers, the square of any real number is always non-negative, so \(x^2 = -25\) has no real solutions.
3. To find solutions, consider the complex number system where \(i\) is defined as the imaginary unit with the property \(i^2 = -1\).
4. Rewrite the equation as \(x^2 = -25 = 25 \times (-1) = 25i^2\), then take the square root of both sides to get \(x = \pm \sqrt{25i^2}\).
5. Simplify the square root to \(x = \pm 5i\), which are the two complex solutions to the equation.

Key Concepts

Solving Quadratic Equations

Quadratic equations are polynomial equations of degree two, typically in the form ax² + bx + c = 0. Solving them involves finding values of x that satisfy the equation, often using factoring, completing the square, or the quadratic formula.

Complex Numbers and Imaginary Unit

When a quadratic equation has no real solutions, such as x² = -25, solutions involve complex numbers. The imaginary unit i is defined as √-1, allowing expressions like √-25 to be written as 5i, representing imaginary solutions.

Square Roots of Negative Numbers

Taking the square root of a negative number is not possible within the real numbers. Instead, it introduces imaginary numbers, where √-a = i√a for a > 0. This concept is essential for understanding solutions to equations like x² = -25.