91Ó°ÊÓ

The 'quadratic equation method' is a classic approach to solving equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a \eq 0\). One common method is by factoring, where we look for two numbers that multiply to give \(ac\) and add up to \(b\). If the quadratic cannot be factored easily, we can use the Quadratic Formula \[x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\]. However, when \(b\) and \(c\) are zero or can be made zero by dividing through by a constant, the equation can be greatly simplified.

For the given exercise, \(3 x^{2}-27=0\), the constant \(b\) is zero, which makes the quadratic method used somewhat different from the usual. Here, we can divide both sides by \(3\) to simplify the equation, making it easier to find the variable \(x\).

Isolating variables is a fundamental skill in algebra. The goal is to get the variable of interest, often \(x\), by itself on one side of the equation. This usually involves performing the same operation on both sides of the equation to keep it balanced.

To apply this concept to our exercise (\(3x^{2}=27\)), we need to isolate \(x^{2}\). We do so by dividing each side of the equation by \(3\), the coefficient of \(x^{2}\). This leads to \(x^{2}=9\), where \(x^{2}\) is isolated and ready to be solved. The idea is to break down the problem into simpler parts to make the solution more evident.

Finding square roots is a process often used to solve quadratic equations. When you have an equation where \(x^{2}\) is isolated, such as \(x^{2}=9\), taking the square root of both sides will help you find the value of \(x\).

It is critical to remember that a squared number has both a positive and a negative root. Therefore, when you take the square root of \(9\), you obtain \(x=3\) and \(x=-3\). Both answers are valid solutions to the original equation. So, finding square roots wraps up the process of solving the quadratic equation and ensures that all possible solutions are accounted for.