Solve each equation in Exercises 15–34 by the square root property. $2x^2-5=-55$

Verified step by step guidance

1. Start by isolating the term with the variable squared. Add 5 to both sides of the equation to get: \(2x^2 - 5 + 5 = -55 + 5\), which simplifies to \(2x^2 = -50\).
2. Next, divide both sides of the equation by 2 to solve for \(x^2\): \(\frac{2x^2}{2} = \frac{-50}{2}\), resulting in \(x^2 = -25\).
3. Apply the square root property, which states that if \(x^2 = k\), then \(x = \pm \sqrt{k}\). So, take the square root of both sides: \(x = \pm \sqrt{-25}\).
4. Recognize that \(\sqrt{-25}\) involves the square root of a negative number, which introduces imaginary numbers. Rewrite \(\sqrt{-25}\) as \(\sqrt{25} \times \sqrt{-1}\).
5. Simplify \(\sqrt{25}\) to 5 and recall that \(\sqrt{-1} = i\), the imaginary unit. Therefore, express the solutions as \(x = \pm 5i\).

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 Quadratic Term

Before applying the square root property, the quadratic term must be isolated on one side of the equation. This involves using algebraic operations such as addition, subtraction, multiplication, or division to rewrite the equation in the form x² = number.

Handling Negative Values Under the Square Root

If the value under the square root (the radicand) is negative, the solutions involve imaginary numbers. Understanding complex numbers and the imaginary unit i (where i² = -1) is essential to express solutions when the radicand is less than zero.