Solving for Two Numbers Given Their Sum and Product: A Systematic Approach
When faced with the problem of finding two numbers x and y where their sum is 15 and their product is 54, a systematic approach is essential. This article will guide you through the process of solving such problems by using algebraic methods, particularly focusing on the utilization of quadratic equations. Whether you're a student, a professional, or simply curious about solving such mathematical challenges, this guide will provide you with a clear and detailed explanation.
Understanding the Given Information
We are given the sum and product of two numbers. Specifically, we have:
The sum of the two numbers is 15: x y 15 The product of the two numbers is 54: xy 54To solve for the values of x and y, we will first express one variable in terms of the other using the sum equation, and then substitute it into the product equation. This will transform the problem into a quadratic equation, which can be solved using the quadratic formula or by factoring.
Expressing One Variable in Terms of the Other
From the sum equation:
x y 15 implies y 15 - x
Substituting and Solving the Quadratic Equation
Next, we substitute the expression for y into the product equation:
x(15 - x) 54
15x - x^2 54
-x^2 15x - 54 0
x^2 - 15x 54 0
This is a quadratic equation in the form of ax^2 bx c 0, where a 1, b -15, and c 54. We can solve this quadratic equation using the quadratic formula:
x frac{-b pm sqrt{b^2 - 4ac}}{2a}
Substituting the values of a, b, and c into the formula:
x frac{15 pm sqrt{(-15)^2 - 4 cdot 1 cdot 54}}{2 cdot 1}
x frac{15 pm sqrt{225 - 216}}{2}
x frac{15 pm sqrt{9}}{2}
x frac{15 pm 3}{2}
This gives us two potential solutions for x:
x frac{18}{2} 9
x frac{12}{2} 6
Now, substituting these back to find y:
When x 9, y 15 - 9 6
When x 6, y 15 - 6 9
Thus, the two numbers are 6 and 9.
Verification
To verify these solutions:
Their sum is indeed 15: 6 9 15 Their product is indeed 54: 6 times 9 54General Approach and Application
This method can be applied to any scenario where the sum and product of two numbers are known. The key steps are to set up the equations, express one variable in terms of the other, substitute, and solve the resulting quadratic equation. By mastering this technique, you can solve similar problems efficiently and accurately.
Conclusion
In conclusion, the systematic approach outlined above provides a clear and methodical way to find two numbers given their sum and product. This technique can be applied to various mathematical problems and is particularly useful for students and professionals in fields such as mathematics, engineering, and science. Whether you are working on problems or simply interested in sharpening your problem-solving skills, this guide should serve as a valuable resource.