Solving the Mathematical Puzzle: One Number is Five More than Another

Have you ever come across a mathematical problem that seems simple at first but requires a bit of algebraic manipulation to solve? This article will walk you through solving a particularly interesting problem, where one number is five more than another. We'll explore the step-by-step approach using quadratic equations, and discuss the importance of properly setting up your problem statements in symbolic form.

Problem Statement

Let's consider a classic problem: one number is five more than another number. Additionally, the square of the smaller number is three more than three times the larger number. What are the numbers?

Solution Approach

To solve this problem, we can follow these steps:

Step 1: Define Variables

Let the smaller number be represented by x and the larger number be represented by y. According to the problem, one number is five more than the other:

y x 5

Step 2: Formulate the Equation

The problem also states that the square of the smaller number is three more than three times the larger number. Using our variable definitions, we can write:

x2 3y 3

Substituting the expression for y from step 1 into the equation:

x2 3(x 5) 3

Step 3: Simplify and Solve

Now, let's simplify and solve the quadratic equation:

x2 3x 15 3

x2 3x 18

x2 - 3x - 18 0

Using the quadratic formula x [-b ± sqrt(b2 - 4ac)] / (2a), where a 1, b -3, and c -18:

x [3 ± sqrt(9 72)] / 2

x [3 ± sqrt(81)] / 2

x [3 ± 9] / 2

x 6 or x -3

Since the square of the smaller number cannot be negative, we discard x -3. Therefore, x 6.

Now, to find the larger number y:

y x 5

y 6 5

y 11

Verification

To verify our solution, we check if the conditions of the problem are satisfied:

x2 62 36

3y 3 3(11) 3 33 3 36

y 6 5 11

The solution is correct.

Conclusion

In conclusion, we have successfully solved a mathematical puzzle using systematic steps, defining variables, and applying algebraic techniques. Understanding how to set up and solve problems symbolically is crucial in mathematics, and this problem serves as an excellent exercise in algebraic manipulation and quadratic equations.