Question

Use the Square Root Method to solve the equation \((x-2)^{2}=9\)

Short answer

The solutions are x = -1 and x = 5.

Step-by-step solution

Isolate the squared term

The equation \[\begin{equation} (x-2)^{2}=9 \end{equation}\] is already isolated.

Take the square root of both sides

Apply the square root to both sides of the equation to remove the square:\[\begin{equation}\tag{2} \sqrt{(x-2)^2} = \pm\sqrt{9}= 3\end{equation}\]This simplifies to \[\begin{equation}\tag{3} x-2 = \pm 3\end{equation}\].

Solve for x

Split the equation into two separate equations:\[\begin{equation}\tag{4a} x-2 = 3\end{equation}\] and\[\begin{equation}\tag{4b} x-2 = -3\end{equation}\].

Solve each equation

Solve the first equation:\[\begin{equation}\tag{5a} x = 3 + 2 = 5\end{equation}\]Solve the second equation:\[\begin{equation}\tag{5b} x = -3 + 2 = -1\end{equation}\].

Key concepts

Solving Quadratic Equations

Quadratic equations are polynomial equations of the form \( ax^2 + bx + c = 0 \). They contain the variable raised to the power of two (squared). When it comes to solving these equations, there are various methods available, one of which is the Square Root Method.

In general, solving quadratic equations can be broken down into three main steps:

• Isolating the squared term: The goal is to get the quadratic term by itself on one side of the equation.


• Taking the square root of both sides: This step simplifies the equation into a more manageable form.


• Solving for the variable: Finally, you solve for the variable by isolating it completely.

Now, let's dive into each step more deeply, using the given example, \((x-2)^2 = 9\).

Isolating the Variable

To solve a quadratic equation using the Square Root Method, the first thing you need to do is isolate the squared term. In the example \((x-2)^2 = 9\), notice that the squared term \((x-2)^2\) is already isolated.

This means we can move on to the next step. However, in cases where the squared term isn't isolated, you might need to rearrange the equation.
Here are some tips for isolating the squared term:

• Move other terms to the opposite side of the equation by adding or subtracting them.


• Divide or multiply both sides if the squared term has a coefficient other than 1.

Once the squared term is isolated, you can proceed to the next step.

Square Root Property

The Square Root Property is a useful tool for solving equations that include squared terms. It tells us that if \((x - a)^2 = b\), then \(x - a = \pm\sqrt{b}\).

In our example, \((x-2)^2 = 9\), we apply the Square Root Property to both sides of the equation:

\( \sqrt{(x-2)^2} = \pm\sqrt{9} \)

Simplifying, we get:

\( x-2 = \pm 3 \)

This gives us two separate linear equations to solve:

\( x - 2 = 3 \)

\( x - 2 = -3 \)
Finally, solving each equation will give us the possible values for \( x \). For the first equation, adding 2 to both sides:

\( x = 5 \)

For the second equation, adding 2 to both sides:

\( x = -1 \)

Therefore, the solutions to the equation \((x-2)^2 = 9\) are \( x = 5 \) and \( x = -1 \).