Problem 15 - Elementary Algebra

Question

Solve the following quadratic equations. \((x+2)^{2}=9\)

Step-by-Step Solution

Answer

The solutions are \( x = 1 \) and \( x = -5 \).

Step 1: Expand the equation

Start by recognizing that \( (x+2)^2 = 9 \) represents a difference of squares. We can solve this by taking the square root of both sides.

Step 2: Take the square root of both sides

Take the square root of both sides to simplify: \[ \sqrt{(x+2)^2} = \sqrt{9} \] This results in \ x+2 = \pm 3 \.

Step 3: Solve for the positive case

First, solve for \( x+2 = 3 \). Subtract 2 from both sides: \ x = 3 - 2 \ Therefore, \ x = 1 \.

Step 4: Solve for the negative case

Next, solve for \( x+2 = -3 \). Subtract 2 from both sides: \ x = -3 - 2 \ Therefore, \ x = -5 \.

Step 5: Verify the solutions

Substitute \( x = 1 \) and \( x = -5 \) back into the original equation to verify. \( (1+2)^2 = 3^2 = 9 \) and \((-5+2)^2 = (-3)^2 = 9\). Both solutions satisfy the original equation.

Key Concepts

square root method

The square root method is a useful approach for solving quadratic equations when the equation is in the form of a perfect square. This method involves taking the square root of both sides of the equation to simplify it.

For example, consider the equation \( (x + 2)^2 = 9 \). Here, the left side of the equation is a perfect square. To solve it using the square root method, follow these steps:

• Take the square root of both sides: \[ \text{\textbackslash sqrt\textbraceleft (x + 2)\textasciicircum2\textbraceright = \text{\textbackslash sqrt\textbraceleft 9\textbraceright} \] This simplifies to \[ x + 2 = \text{\pm} 3 \]

You can break this into two separate equations:

• \( x + 2 = 3 \)
• \( x + 2 = -3 \)

Finally, solve each equation for \( x \) by isolating the variable. By doing so, you will find the two possible solutions for the quadratic equation.

difference of squares

The term 'difference of squares' refers to a special way of recognizing and solving quadratic equations. This method is most useful for equations that can be expressed as the difference between two perfect squares. However, in the provided exercise, recognizing the form is key to applying the square root method.

In the equation \( (x + 2)^2 = 9 \), observe that both sides are squares: the left side is a squared binomial and the right side is simply a number squared (\(3^2\)).

When an equation is structured this way, it hints at the use of the square root method. Essentially, the difference of squares concept helps identify the simpler path to solve the equation, leading directly to breaking down the problem via square roots.

solution verification

Verifying your solutions is a critical step in solving quadratic equations. It ensures that the solutions you have found actually satisfy the original equation.

For the provided exercise, the solutions found are \( x = 1 \) and \( x = -5 \). To verify:

1. Substitute \( x = 1 \) back into the original equation \( (x + 2)^2 = 9 \):

• \( (1 + 2)^2 = 3^2 = 9 \)
• This verifies that \( x = 1 \) is a correct solution.

2. Substitute \( x = -5 \) back into the original equation:

• \( (-5 + 2)^2 = (-3)^2 = 9 \)
• This verifies that \( x = -5 \) is also a correct solution.

By checking that both solutions satisfy the original equation, you can be confident in their correctness.