Problem 32 - Beginning and Intermediate Algebra

Question

Solve using the square root property. Simplify all radicals. $$ w^{2}=0.49 $$

Step-by-Step Solution

Answer

The solutions are \( w = 0.7 \) and \( w = -0.7 \).

Step 1: Write the equation

The given equation is \[ w^2 = 0.49 \]. The goal is to solve for \( w \).

Step 2: Apply the Square Root Property

According to the square root property, if \( w^2 = c \), then \( w = \pm \sqrt{c} \). Applying it to the given equation: \[ w = \pm \sqrt{0.49} \].

Step 3: Simplify the Square Root

Solve for \( \sqrt{0.49} \). Since \( 0.49 = \(0.7\)^2 \), we find that \[ \sqrt{0.49} = 0.7 \].

Step 4: Find the Solutions

Considering the positive and negative roots, the solutions are \[ w = \pm 0.7 \].

Key Concepts

Solving Quadratic Equations

When you encounter a quadratic equation of the form \( w^2 = c \), your primary goal is to solve for the variable, in this case, \( w \). The Square Root Property is a powerful tool for these types of equations. Here’s how it works:
According to the Square Root Property, if you have an equation like \( w^2 = c \), then \( w = \pm \sqrt{c} \). Notice the \( \pm \) symbol; it signifies both the positive and negative roots of the square root.
Let’s apply this to our given equation, \( w^2 = 0.49 \). By applying the Square Root Property, we get:
\( w = \pm \sqrt{0.49} \).
This step helps break down the equation so we can move on to simplifying the square root.

Simplifying Radicals

Simplifying radicals is the next crucial step. In our example, we need to simplify \( \sqrt{0.49} \). This simplification involves finding out what number multiplied by itself gives you 0.49.
By recognizing that \( 0.49 = (0.7)^2 \), we know that:
\( \sqrt{0.49} = 0.7 \).
Here’s a pro tip: if the number under the square root is a perfect square, like 0.49 (also 7 times 7), it makes our job easier. Simplified roots lead to simpler answers, which brings us to our final solutions.

Positive and Negative Roots

When we solve quadratic equations using the Square Root Property, we always consider both positive and negative roots. This is represented by the \( \pm \) symbol.
In our example, after simplifying \( \sqrt{0.49} = 0.7 \), the final solutions are:
\( w = \pm 0.7 \), meaning \( w = 0.7 \) and \( w = -0.7 \).
Why do we consider both roots? Because a negative number squared also results in a positive number (for example, \( (-0.7)^2 = 0.49 \)).
Therefore, it’s essential to include both the positive and negative solutions every time you solve a quadratic equation by using the square root property.