91Ó°ÊÓ

Quadratic equations, which often appear in the form of \(ax^2 + bx + c = 0\), are fundamental in algebra. Solving these types of equations involves finding the values of \(x\) that make the equation true. There are multiple methods to tackle quadratic equations, including factoring, completing the square, using the quadratic formula, and the method in our exercise, the square root property.

When the quadratic equation can be rewritten so that a perfect square is equal to a constant (such as \(a(x+h)^2 = k\)), the square root property becomes particularly useful. This property states that if \(a(x+h)^2 = k\), then \(x+h = \pm\frac{\sqrt{k}}{\sqrt{a}}\) or \(x+h = -\pm\frac{\sqrt{k}}{\sqrt{a}}\). Afterward, \(x\) can be isolated to find the solution(s). In the provided exercise, \(3(x+4)^2=21\) simplifies directly to a form suitable for the application of the square root property, putting us one step closer to finding the values of \(x\).

The first step to effectively use the square root method is to isolate the squared term. Isolating the squared term means getting the square term by itself on one side of the equation. In the given example, \(3(x+4)^2=21\), we isolate \( (x+4)^2 \) by dividing both sides of the equation by 3. This process simplifies the problem so we are dealing with \( (x+4)^2 \) without any additional coefficients in front of it.

It's important to simplify the equation as much as possible before moving on to the next step. In this case, we end up with \( (x+4)^2 = 7 \), a much neater expression to work with, showcasing why isolating the square is a critical move toward solving quadratic equations.

Once we have the squared term by itself as in \( (x+4)^2 = 7 \), we can apply the square root method. This technique takes advantage of the fact that the square root of a square number, or a number raised to the power of two, is the base of that term. Therefore, taking the square root of both sides of an equation will 'undo' the squaring.

To apply this method, we need to remember that there are always two possible square roots for any positive number: one positive and one negative. For this reason, finding the square root of both sides of our simplified equation gives us \( x+4 = \pm\sqrt{7} \), introducing a plus/minus symbol \( (\pm) \) to account for both possibilities. The final step is to solve for \( x \) and accept the two solutions, positive and negative, as \( x = -4 \pm \sqrt{7} \).