Solve each equation using the square root property. x² = -400

Verified step by step guidance

1. Recognize that the equation is in the form \(x^2 = c\), where \(c\) is a constant. The square root property states that if \(x^2 = c\), then \(x = \pm \sqrt{c}\).
2. Identify the constant on the right side of the equation: here, \(c = -400\).
3. Since \(c\) is negative, recall that the square root of a negative number involves imaginary numbers. Express \(-400\) as \(-1 \times 400\) to separate the negative sign.
4. Rewrite the square root using imaginary unit \(i\), where \(i = \sqrt{-1}\), so \(\sqrt{-400} = \sqrt{-1 \times 400} = \sqrt{-1} \times \sqrt{400} = i \times 20\).
5. Apply the square root property to write the solutions as \(x = \pm 20i\).

Key Concepts

Square Root Property

The square root property states that if x² = k, then x = ±√k. This method is used to solve quadratic equations that can be written in the form x² = a constant. It involves taking the square root of both sides, remembering to include both positive and negative roots.

Complex Numbers and Imaginary Unit

When solving equations like x² = -400, the square root of a negative number involves imaginary numbers. The imaginary unit i is defined as √(-1), allowing us to express roots of negative numbers as multiples of i, such as √(-400) = 20i.

Solving Quadratic Equations

Quadratic equations can be solved by isolating the squared term and applying the square root property. Understanding how to manipulate the equation and interpret solutions, including real and complex roots, is essential for correctly solving and interpreting results.