91Ó°ÊÓ

Quadratic equations are polynomials that follow the general form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). Solving a quadratic equation means finding the values of \(x\) that make the equation true. The square root method is one of the strategies applied when the equation can be simplified to the form \(x^2 = d\), which is an especially convenient format. This approach avoids more complex methods such as factoring or using the quadratic formula. It is most effective when the quadratic equation has no \(bx\) term and is easily simplified to isolate \(x^2\) on one side of the equation.

For example, in our given problem, by dividing both sides by 3, the quadratic equation \(3x^2 = 27\) becomes \(x^2 = 9\), which is a form readily suitable for the square root method. A key feature of quadratic equations is that they can have two real solutions—this is exactly what we'll exploit using the square root method, which will lead us to the next concept of isolating the variable.

Isolating the variable is a fundamental skill in algebra that involves manipulating the equation so that the unknown variable, often \(x\), is by itself on one side of the equation. The goal is to rewrite the equation such that it becomes \(x = \textrm{{something}}\) or \(x^2= \textrm{{something}}\) in the case of a quadratic equation. This process includes performing the same mathematical operation on both sides of the equation to maintain equality.

In the provided exercise, \(3x^2 = 27\), we isolate \(x^2\) by dividing both sides of the equation by 3, which gives us \(x^2 = 9\). This step is crucial because it paves the way for the extraction of square roots, which will be the next stage. When isolating the variable, it's important to remember that whatever operation you do to one side of the equation, you must do to the other side.

Extracting square roots is a technique used to solve equations where the variable is squared, such as \(x^2 = d\). It involves taking the positive and negative square roots of both sides of the equation to find all possible solutions for the variable. Mathematically, to extract the square root of \(x^2\), you write \(x = \pm\sqrt{d}\). The 'plus-minus' symbol (\(\pm\)) signifies that there will be two solutions: one positive and one negative.

Referring to the exercise \(x^2 = 9\), we take the square root of both sides to obtain \(x = \pm3\). The reason we take both the positive and the negative square roots is that both \(3^2\) and \( (-3)^2\) equal to 9. It is a fundamental property of squares that a squared negative number yields a positive result. Therefore, it is important to consider both solutions in the context of the original equation to determine if they are valid.