Problem 9 - Elementary and Intermediate Algebra

Question

For each number, find all of its square roots. $$ 49 $$

Step-by-Step Solution

Answer

7 and -7

Step 1: Understand the Problem

The task is to find the square roots of the number 49. A square root of a number is a value that, when multiplied by itself, gives the original number.

Step 2: Formulate the Equation

Set up the equation for finding the square roots: \[ x^2 = 49 \]This equation states that we need to find a number \( x \) such that \( x \) squared (multiplied by itself) equals 49.

Step 3: Solve for Positive Square Root

Take the positive square root of 49: \[ x = \sqrt{49} \]This gives: \[ x = 7 \]Thus, one square root of 49 is 7.

Step 4: Solve for Negative Square Root

Take the negative square root of 49: \[ x = -\sqrt{49} \]This gives: \[ x = -7 \]Thus, the other square root of 49 is -7.

Step 5: List All Square Roots

Combine the results from the previous steps: The square roots of 49 are 7 and -7.

Key Concepts

positive square root

A 'positive square root' is the non-negative root of a number. When finding the square root of a positive number, we look for a value that, when multiplied by itself, returns the original number. For example, the positive square root of 49 is found as follows:

\( \sqrt{49} = 7 \).

This means that 7 is a positive square root of 49 because 7 times 7 equals 49.

In general, \( \sqrt{x} \) refers to the positive square root. Positive square roots are part of many mathematical solutions and are always greater than or equal to zero.

negative square root

The 'negative square root' of a number is the negative value that, when multiplied by itself, also returns the original number. To find this, we simply take the negative of the positive square root. For instance, the negative square root of 49 is:

\[ -\sqrt{49} = -7 \]

Thus, -7 is also a square root of 49 because multiplying -7 with itself (\( -7 \times -7 \)) gives 49.

Square roots can be both positive and negative because \( (-a)^2 \) is equal to \( a^2 \). Therefore, anytime you encounter a square root problem, remember to consider both positive and negative roots.

solving equations

Solving equations involving square roots involves understanding the relationship between a number and its roots. The equation \( x^2 = 49 \) exemplifies this. To solve for x, follow these steps:

• Recognize that any square root problem essentially asks, 'Which numbers, when squared, will give the original number?'
• Set up the equation: In this case, \( x^2 = 49 \).
• Identify both the positive and negative solutions.

This leads to two possible values for x:

\ \[ x = \sqrt{49} = 7 \ x = -\sqrt{49} = -7 \]

Thus, x can be either 7 or -7. By systematically approaching the equation and exploring both the positive and negative solutions, you can confidently solve problems involving square roots.