Introduction to the Problem

This article explores a common mathematical problem where we need to find two numbers that have a specific sum and product. The problem presented here is to find two numbers whose sum is 22 and whose product is 105. We will solve this using both algebraic methods and the quadratic formula, ensuring a clear and step-by-step explanation for better comprehension.

Algebraic Approach to Solving the Problem

Let's denote the two numbers as (x) and (y). We know that:

Equation 1: (x y 22)

Equation 2: (xy 105)

From Equation 1, we can express (y) in terms of (x): [ y 22 - x ]

Substitute (y 22 - x) into Equation 2:

[ x(22 - x) 105 ]

Expanding and rearranging the equation, we get:

[ 22x - x^2 105 ]

[ x^2 - 22x 105 0 ]

This is a quadratic equation, which can be solved using the quadratic formula:

[ x frac{-b pm sqrt{b^2 - 4ac}}{2a} ]

For the given equation (x^2 - 22x 105 0), the coefficients are (a 1), (b -22), and (c 105). Substituting these into the quadratic formula gives us:

[ x frac{22 pm sqrt{(-22)^2 - 4 cdot 1 cdot 105}}{2 cdot 1} ]

[ x frac{22 pm sqrt{484 - 420}}{2} ]

[ x frac{22 pm sqrt{64}}{2} ]

[ x frac{22 pm 8}{2} ]

This yields two solutions for (x): [ x frac{30}{2} 15 ] [ x frac{14}{2} 7 ]

Thus, the two numbers are 15 and 7. Verify these solutions:

Therefore, the two numbers that have a sum of 22 and a product of 105 are 15 and 7.

Alternative Method Using Algebraic Manipulation

Another approach can be by directly substituting and solving:

Let's denote the two numbers as (x) and (y). The given conditions are:

(x y 22)

(xy 105)

From (x y 22), we can express (y) in terms of (x): [ y 22 - x ]

Substitute (y 22 - x) into (xy 105): [ x(22 - x) 105 ]

Expanding and rearranging, we get the same quadratic equation:

[ x^2 - 22x 105 0 ]

Using the same quadratic formula, we get:

[ x frac{22 pm 8}{2} ]

[ x 7 quad text{or} quad x 15 ]

Therefore, the numbers are 15 and 7, and the pairs can also be 7 and 15.

Observational Approach

A third, more intuitive approach can be observed directly by inspection:

From (15 times 7 105) and adding them together, we get:

(15 7 22)

(15 times 7 105)

This confirms that 15 and 7 are the correct numbers.

Conclusion

Both algebraic and observational methods confirm that the numbers satisfying the conditions of having a sum of 22 and a product of 105 are 15 and 7. This article provides a clear, step-by-step solution to the problem, ensuring a comprehensive understanding of the underlying mathematical concepts.