Problem 47 - Elementary Algebra

Question

Solve using the Square Root Property. \((x+5)^{2}=4\)

Step-by-Step Solution

Answer

The solutions are \( x = -3 \) and \( x = -7 \).

Step 1 - Isolate the squared term

Identify the term that is squared in the equation. In this case, \( (x+5)^{2} \) is already isolated on one side of the equation.

Step 2 - Take the square root of both sides

Apply the square root to both sides of the equation to eliminate the square. \[ (x+5) = \pm\sqrt{4} \] Remember that taking the square root introduces both a positive and a negative solution.

Step 3 - Simplify the square root

Simplify \( \sqrt{4} \), which is equal to 2. Therefore, the equation \[ x+5 = \pm 2 \] becomes \[ x+5 = 2 \] and \[ x+5 = -2 \]

Step 4 - Solve for x

Solve both equations for \( x \). For \( x + 5 = 2 \), subtract 5 from both sides to get \( x = -3 \). For \( x + 5 = -2 \), subtract 5 from both sides to get \( x = -7 \).

Step 5 - State the solutions

The solutions to the equation \( (x+5)^{2}=4 \) are \( x = -3 \) and \( x = -7 \).

Key Concepts

solving quadratic equations

To solve quadratic equations, we often look for methods that simplify the process. One such method is the Square Root Property. Quadratic equations are in the form \[ ax^2 + bx + c = 0 \], where 'a', 'b', and 'c' are constants. Various methods to solve these include: factoring, using the quadratic formula, completing the square, or utilizing the Square Root Property.

Today we focus on solving using the Square Root Property, especially useful when the equation can be written in the form \[ (ax + b)^2 = c \]. This allows us to apply the property and simplify our work.

square root property

The Square Root Property is a powerful tool for solving equations with perfect squares. It states:

• If \[ a^2 = b \], then \[ a = \text{±}\text{√}b \].

This property is incredibly useful because it allows us to directly simplify and solve for the variable inside the square.

In our example from the exercise, \[ (x+5)^2 = 4 \], the squared term is isolated. We can apply the Square Root Property to get:
\[ x + 5 = \text{±}\text{√}4 \].

isolating the squared term

Isolating the squared term is the first critical step when solving using the Square Root Property.

• Make sure the term involving the square is alone on one side of the equation.

In our example:
\[ (x + 5)^2 = 4 \],
the squared term \[ (x + 5)^2 \] is already isolated. No extra steps are needed here. If it wasn't isolated, you'd need to perform operations to move other terms to the other side of the equal sign first.

simplifying square roots

After isolating the squared term and applying the Square Root Property, the next step is to simplify the square root.

• Calculate the square root of the value, remembering both positive and negative roots.

In our example: \[ x + 5 = \text{±}\text{√}4 \] simplifies to: \[ x + 5 = 2 \] and \[ x + 5 = -2 \].
Simplifying further, we solve for \[ x \] in both equations by subtracting 5:
\[ x = -3 \] and \[ x = -7 \]. Thus, our solutions are \[ x = -3 \] and \[ x = -7 \].