Solve each equation using the square root property. See Example 2. 48 - x^2 = 0

Verified step by step guidance

1. Start by isolating the term with the variable. Move the \(x^2\) term to the other side of the equation: \(48 = x^2\).
2. Apply the square root property, which states that if \(x^2 = a\), then \(x = \pm \sqrt{a}\).
3. Take the square root of both sides of the equation: \(x = \pm \sqrt{48}\).
4. Simplify the square root if possible. Break down \(\sqrt{48}\) into its prime factors: \(\sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3}\).
5. Since \(\sqrt{16} = 4\), the expression simplifies to \(x = \pm 4\sqrt{3}\).

Key Concepts

Square Root Property

The square root property states that if an equation is in the form x^2 = k, where k is a non-negative number, then the solutions for x can be found by taking the square root of k. This results in two possible solutions: x = √k and x = -√k. This property is essential for solving quadratic equations that can be rearranged into this standard form.

Rearranging Equations

Rearranging equations involves manipulating the equation to isolate the variable of interest. In the context of the given problem, we first need to move all terms to one side to set the equation to zero. This step is crucial for applying the square root property effectively, as it allows us to express the equation in the standard form required for solving.

Quadratic Equations

Quadratic equations are polynomial equations of the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. They can be solved using various methods, including factoring, completing the square, and applying the square root property. Understanding the nature of quadratic equations is vital for recognizing when and how to apply different solving techniques.