Match the equation in Column I with its solution(s) in Column II. x² + 5 = 0

Verified step by step guidance

1. Start with the given equation: \(x^2 + 5 = 0\).
2. Isolate the \(x^2\) term by subtracting 5 from both sides: \(x^2 = -5\).
3. Recognize that \(x^2 = -5\) has no real solutions because the square of a real number cannot be negative.
4. To find the solutions, take the square root of both sides, remembering to include both the positive and negative roots: \(x = \pm \sqrt{-5}\).
5. Express the square root of a negative number using imaginary numbers: \(x = \pm \sqrt{5}i\), where \(i\) is the imaginary unit with the property \(i^2 = -1\).

Key Concepts

Solving Quadratic Equations

A quadratic equation is a polynomial equation of degree two, typically in the form ax² + bx + c = 0. Solving it involves finding values of x that satisfy the equation, often by factoring, completing the square, or using the quadratic formula.

Imaginary and Complex Numbers

When a quadratic equation has no real solutions (e.g., when the discriminant is negative), solutions involve imaginary numbers. The imaginary unit i is defined as √-1, allowing solutions to be expressed as complex numbers with real and imaginary parts.

Discriminant and Nature of Roots

The discriminant (Δ = b² - 4ac) determines the nature of the roots of a quadratic equation. If Δ < 0, the equation has two complex conjugate solutions; if Δ = 0, one real repeated root; if Δ > 0, two distinct real roots.