91Ó°ÊÓ

The quadratic formula is a powerful tool in algebra that provides a method to solve any quadratic equation of the form \(ax^2 + bx + c = 0\). When applying the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), we calculate the values of \(x\) that make the equation true. It neatly encapsulates the process of finding the roots of a quadratic equation into a single, straightforward expression.

To properly use the quadratic formula, one must identify the coefficients \(a\), \(b\), and \(c\) from the quadratic equation. In the exercise given, the equation was simplified to \(x^2 - 5 = 0\), indicating that \(a = 1\), \(b = 0\), and \(c = -5\). Substituting these values into the formula provides us with the solutions for \(x\).

Rearranging equations is a fundamental skill in algebra that involves manipulating an equation to isolate the variable of interest on one side. This step is usually performed to prepare the equation for further solution methods, such as factoring or using the quadratic formula.

In the exercise provided, the initial step involved moving all terms with the variable \(x\) to one side of the equality and the constants to the other side. This was achieved by subtracting \(4x^2\) from both sides, which left us with \(x^2 - 5 = 0\). The process of rearranging is essential, as it simplifies the equation and can uncover the structure needed to apply specific algebraic methods for finding the variable's value.

Radicals—or root expressions—play a significant role in algebra, particularly when dealing with quadratic equations. They are symbols used to indicate the root of a number or an expression. The most common radical in algebra is the square root, denoted by \(\sqrt{ }\).

When solving quadratic equations, finding the square root of a number is often necessary, especially when applying the quadratic formula. The presence of the square root sign in the formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) arises from completing the square in the derivation of the quadratic formula. The solutions to our exercise, \( x = \sqrt{5} \) and \( x = -\sqrt{5} \), illustrate how radicals are used to express the solutions to equations where the variable is squared.