Problem 34 - Elementary Algebra

Question

For the following problems, find the two square roots of the given number. $$ 144 $$

Step-by-Step Solution

Answer

Answer: The two square roots of 144 are ±12, which include the positive square root 12 and the negative square root -12.

Step 1: Identify the given number

We are given the number 144, and our task is to find its two square roots.

Step 2: Take the square root of the given number

We will now find the square root of 144. It's essential to remember that a given positive number always has two square roots: one positive and one negative. $$ \sqrt{144} = \pm 12 $$ In this case, the positive square root of 144 is 12.

Step 3: Find the negative square root of the number

As mentioned earlier, every positive number has two square roots, one positive and one negative. Since we found the positive square root in the previous step, we can easily find the negative square root. $$ -\sqrt{144} = -12 $$ So, the negative square root of 144 is -12.

Step 4: Write the final answer

Now that we have found both the square roots of 144, we can express our final answer. The two square roots of 144 are: $$ \pm 12 $$

Key Concepts

Positive Square Root

When you think about square roots, the positive square root is usually what comes to mind first. A positive square root of a number is that value which, when multiplied by itself, gives the original number. For instance, the positive square root of 144 is 12 because 12 multiplied by 12 results in 144.

It's important to note, though, that while some think of the square root symbol \( \sqrt{} \) as having only a positive value, it can technically represent both positive and negative roots. However, by convention, when you see \( \sqrt{144} \) or a similar expression without any sign, the expression refers to the positive square root only.

Understanding positive square roots is fundamental when you're learning to solve quadratic equations, dealing with geometry involving areas, or working out scientific and engineering problems that require precise calculations. Simply put, positive square roots are everywhere in mathematics!

Negative Square Root

Though the concept that a square root can also be negative might seem counterintuitive at first, it's just as important as its positive counterpart. The negative square root is simply the negative of the positive square root. For example, the negative square root of 144 is -12 because when you multiply -12 by itself, you also get 144.

Mathematically, while the square of both 12 and -12 yield the same positive result, having a negative square root is essential for depicting a complete mathematical solution when paired with the positive one.

• If \( x^2 = 144 \), then \( x = 12 \) or \( x = -12 \).
• It’s just a simple way to show there are two possible solutions.

In practical terms, although negative square roots do not apply to measurements of physical quantities (lengths, areas) as they can't be negative, they are essential to consider when solving algebraic equations and other mathematical problems where variables can take negative values.

Identifying Square Roots

Identifying square roots of any number is a process that involves recognizing both the positive and negative roots of a given number.

• Start by determining a number that, when multiplied by itself, equals the given number.
• This will provide the positive square root.
• The negative square root is simply the negative version of this number.

Take the example of 144:
- The positive square root is \( +12 \) since \( 12 \times 12 = 144 \).
- The negative square root is \( -12 \) because \( (-12) \times (-12) = 144 \).

Using this approach helps ensure you consider all possibilities when working with square roots. It is a particularly useful skill when solving quadratic equations, as these often require you to identify both square roots of a number to find a complete solution. Identifying square roots correctly opens the door to more complex mathematical operations and ensures comprehensive solutions to algebraic problems.