Solving a Complex Mathematical Problem: The Square Root Relates to a Product

Mathematics often presents us with complex problems that require a systematic approach to solve. One such problem involves the relationship between a square root and a product. Let's explore a specific example where the square root of a number is three more than twice the product of two and three. We will break down the problem step by step to reach a solution.

Step-by-Step Breakdown

The problem statement is:

The square root of a number is three more than twice the product of two and three. What is the number?

Step 1: Calculate the Product of Two and Three

We need to start by finding the product of two and three:

2 times; 3 6

Step 2: Find Twice the Product

Next, we calculate twice the product of two and three:

2 times; 6 12

Step 3: Add Three to This Result

Now, we add three to the result from the previous step:

12 3 15

Step 4: Set Up the Equation

According to the problem, the square root of the number we are looking for is equal to 15. Thus, we can set up the following equation:

sqrt{x} 15

Step 5: Solve for x

To solve for the unknown number, we square both sides of the equation:

x 15^2 225

Therefore, the number is 225.

Verification

We can verify our result by plugging the number back into the original equation:

sqrt{225} 15

This confirms that 225 is indeed the correct answer.

Alternative Example

Problem: Find a Number Where the Square Root Equals 15 Times a Different Product

Let's consider another similar problem involving a different product:

Premises: (sqrt{r} 22 times; 33)

Assumptions:

Let r the number

Let (sqrt{r} sqrt{r})

Step-by-Step Solution

We need to solve the equation step by step as follows:

Square both sides of the equation to simplify it:

(left[sqrt{r} 22 times; 33right]^2)

(r (22 times; 33)^2)

Calculate the product:

22 times; 33 726

(r 726^2)

Further simplification:

726^2 527076

Upon verification, we find that the number is 527076.

Another Example

Problem: Another Expression Involving a Square Root

Consider this simpler example:

Premises: $r81$

Cross-Checking: Let's verify if the number 81 satisfies the condition:

x^2 81

Therefore, x 9

Checking the product: 3 times; 2 times; 3 18

The square root of 81 is 9, which is 18 - 9 9. This confirms the relationship.

Conclusion

Understanding and solving complex mathematical problems requires a clear approach and attention to detail. In this article, we have explored how to relate a square root to a product in a step-by-step manner. By breaking down the problem into smaller, manageable steps, we can systematically resolve even the most complex equations. This methodology is not only useful for academic purposes but also practical in various real-world applications involving mathematics.