Problem 1 - College Algebra with Modeling and Visualization

Question

Find the square roots of the number. Approximate your answers to the nearest hundredth whenever appropriate. $$25$$

Step-by-Step Solution

Answer

The square roots of 25 are 5 and -5.

Step 1: Understanding Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, if \( x^2 = 25 \), then \( x \) is the square root of 25.

Step 2: Identify the Positive Square Root

To find the square root of 25, identify the possible integer value that squares to give 25. We know that 5 times 5 equals 25, so \( \sqrt{25} = 5 \).

Step 3: Identify the Negative Square Root

Square roots can be both positive and negative because both positive and negative values squared will result in a positive number. Therefore, \( -5 \times -5 = 25 \). Hence, \( \sqrt{25} = -5 \) is also a valid square root.

Step 4: Conclude with Both Square Roots

The two square roots of 25 are \( 5 \) and \( -5 \). Since these are integers, they do not need any rounding.

Key Concepts

Positive Square Root

When we talk about square roots, the most common one we refer to is the positive square root. It is often called the "principal square root." Given the number 25, let's focus on finding this particular value.

The definition is straightforward: the positive square root is a number that, when multiplied by itself, results in the original number. For instance, with 25, if you think of a number that when squared (multiplied by itself) equals 25, you immediately come to 5. That's because:

• 5 times 5 equals 25

Therefore, 5 is the positive square root of 25. We write this as \( \sqrt{25} = 5 \).

This value is unique for non-negative numbers and is typically what you get from calculators or when you see the square root symbol. It’s important to remember this for comparison with the negative square root.

Negative Square Root

In addition to a positive square root, every positive number also has a negative square root. This can be a bit confusing at first, but it becomes clear when you understand the properties of multiplication.

Consider our number 25 again. While we already established that multiplying 5 by itself gives 25, consider what happens if we multiply a negative number by itself:

• \(-5\) times \(-5\) also equals 25

This is because when you multiply two negative numbers, the negatives cancel each other out to produce a positive product. Hence, \(-5\) is the negative square root of 25, written as \(\sqrt{25} = -5\).

The negative square root is not as commonly referenced as the positive one, but it is equally valid. It's essential to understand that while the symbol \(\sqrt{}\) commonly indicates the positive root, square root expressions can imply both a positive and negative solution.

Integer Solutions

Integer solutions are calculations where the results are whole numbers, without fractions or decimals. When dealing with square roots, especially in textbook problems, integer solutions are quite common, as seen with our number 25.

Here, 25 has two integer solutions: 5 and -5.

• Both of these solutions are integers because they are whole numbers.
• They perfectly satisfy the conditions for square roots as both 5 and -5, when squared, result in 25.

People often prefer integer solutions because they are neat and straightforward. Unlike non-perfect square roots, such as \(\sqrt{2}\), which result in irrational numbers, perfect squares like 25 lead to integer answers that are easy to work with.