Problem 24 - Algebra 1

Question

Find all square roots of the number or write no square roots. Check the results by squaring each root. $$144$$

Step-by-Step Solution

Answer

The square roots of 144 are 12 and -12.

Step 1: Identify the number to find its square roots

The number provided in the exercise is 144.

Step 2: Calculate the Square Roots

To find the square roots of this given number, it is needed to calculate the positive and the negative square roots separately. \nFor positive root, \(\sqrt{144}=12\) and, for the negative root, \(-\sqrt{144}=-12\). Thus the square roots of 144 are 12 and -12.

Step 3: Verification

Check the results by squaring each root. Indeed, \(12^2=144\) and \((-12)^2=144\). Hence, both roots are indeed the square roots of 144.

Key Concepts

Positive Square Root

Finding the positive square root of a number is like asking what number, when multiplied by itself, gives you the original number. For the number 144, the positive square root is quite simple to find. It is 12. Why 12 and not some other number? Because 12 times 12 is 144!

When we express this in mathematics, we write it as \(\sqrt{144} = 12\). The symbol \(\sqrt{\ }\) is called the square root symbol. Always remember, when you just see this symbol without a negative sign, it refers to the positive square root.

Square roots are powerful tools, especially in geometry and algebra, and understanding their positive values is fundamental to many problems. Knowing how to find and interpret the positive square roots of numbers provides you with a basic skill that will surface time and again in mathematics.

• Positive square roots are always non-negative.
• They are often used in real-world calculations where negative values do not make sense.

Negative Square Root

Now, let's talk about the other sibling in the square root family—the negative square root. You might be wondering, isn't the square root supposed to be positive? Actually, each positive number has two square roots: one positive and one negative.

The negative square root of 144 is -12. This is shown as \(-\sqrt{144} = -12\). It’s basically taking the positive square root and adding a negative sign in front. When you multiply \(-12\) by \(-12\), you get back to 144, because a negative times a negative equals a positive.

You might not often see the negative square root in everyday measurements or distances, but it's vital in algebra and other mathematical scenarios where both negative and positive values shed light on solutions.

• The negative square root is simply the negative of the positive square root.
• Both square roots satisfy the equation \(x^2=144\).

Verification of Results

Verification is an essential step to ensure that your calculated square roots are correct. Imagine solving a puzzle; you always double-check to see if the pieces fit!

To verify square roots, what you do is square both solutions and see if they lead back to the original number. For the number 144 with roots 12 and -12, let's check:

• \(12^2 = 12 \times 12 = 144\)
• \((-12)^2 = (-12) \times (-12) = 144\)

Both results bring us back to 144, confirming our roots are correct. Always remember this verification step, as it'll save you from potential errors down the line.

Verification not only confirms your current answer but also reinforces understanding, helping make square roots and their properties a more intuitive part of your mathematical toolkit.

• Always verify by squaring your results.
• Correct verification will end in arriving back at the original number.