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Finding Two Numbers with a Sum of 27 and a Product of 182

Unveiling the Mystery: Finding Two Numbers with a Sum of 27 and a Product of 182 ,In the realm of mathematics, puzzles often come in the form of equations waiting to be solved. One such puzzle involves finding two numbers that satisfy the unique conditions of having a sum of 27 and a product of 182.

The challenge lies not only in deciphering the values of these numbers but also in understanding the underlying principles that govern their relationship. This article will embark on a journey to unravel this mathematical enigma and shed light on the methods employed to solve it.

Understanding the Problem

The problem presents us with two key pieces of information: the sum of the two numbers is 27, and their product is 182. Let the two numbers be represented by x and y. Translating the given information into equations, we have:

  1. x + y = 27
  2. x * y = 182

Solving the Equations

The next step is to solve this system of equations to determine the values of x and y. There are several methods to approach this, including substitution, elimination, or even using specialized calculators or software. For the sake of clarity, let’s use the substitution method.

From equation 1, we can isolate one variable and substitute it into equation 2 to solve for the other variable. Let’s solve for y in equation 1:

x + y = 27 y = 27 – x

Substitute this value of y into equation 2:

x * y = 182 x * (27 – x) = 182 27x – x^2 = 182

Rearrange the equation:

x^2 – 27x + 182 = 0

Factoring or Using the Quadratic Formula

Unveiling the Mystery: Finding Two Numbers with a Sum of 27 and a Product of 182 ,At this point, we have a quadratic equation that can be solved either by factoring or by using the quadratic formula. Factoring might not yield nice integer solutions in this case, so let’s use the quadratic formula:

x = (-b ± √(b^2 – 4ac)) / 2a

For our equation, a = 1, b = -27, and c = 182:

x = (27 ± √((-27)^2 – 4 * 1 * 182)) / (2 * 1) x = (27 ± √(729 – 728)) / 2 x = (27 ± √1) / 2

This gives us two possible solutions for x:

  1. x = (27 + 1) / 2 = 14
  2. x = (27 – 1) / 2 = 13

Now that we have the values of x, we can use the equation y = 27 – x to find the corresponding values of y:

  1. x = 14 → y = 27 – 14 = 13
  2. x = 13 → y = 27 – 13 = 14

Conclusion

In conclusion, the mystery of finding two numbers with a sum of 27 and a product of 182 has been successfully unraveled. The numbers are 13 and 14, satisfying both the sum and product conditions. Through algebraic manipulation and mathematical reasoning, we’ve illuminated the process of solving such a problem. This endeavor underscores the elegance and power of mathematics in deciphering even the most perplexing puzzles.

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