Solve each equation by the square root property. $\frac{x^2}{2}+5=-3$

Verified step by step guidance

1. Start with the given equation: \(\frac{x^2}{2} + 5 = -3\).
2. Isolate the term containing \(x^2\) by subtracting 5 from both sides: \(\frac{x^2}{2} = -3 - 5\).
3. Simplify the right side: \(\frac{x^2}{2} = -8\).
4. Eliminate the fraction by multiplying both sides of the equation by 2: \(x^2 = -16\).
5. Apply the square root property: take the square root of both sides, remembering to include both the positive and negative roots: \(x = \pm \sqrt{-16}\).

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 simplifies solving by isolating the squared term and then taking the square root of both sides.

Isolating the Variable Term

Before applying the square root property, the equation must be manipulated to isolate the squared term on one side. This involves using algebraic operations such as addition, subtraction, multiplication, or division to simplify the equation and prepare it for taking square roots.

Handling Negative Values Under the Square Root

If the value under the square root (the radicand) is negative, the solutions involve imaginary or complex numbers. Recognizing when the radicand is negative is important, as it indicates no real solutions exist, and the answer must be expressed using the imaginary unit i, where i² = -1.