Solving for Numbers Given Their Product and Sum

Today, we delve into a problem where the product of two numbers is given, and the sum of the numbers is also specified. This problem requires us to find the actual values of these numbers. Let's explore this through an interesting algebraic approach.

Problem Statement

The product of two numbers is 5, and their sum is 9/2. We need to find these two numbers.

Algebraic Solution

Let's denote the two numbers as y.

We are given two pieces of information:

The product of the two numbers is 5: xy 5 The sum of the two numbers is 9/2: x y frac{9}{2}

From the second equation, we can express one of the variables in terms of the other:

y frac{9}{2} - x

Substituting this expression for y into the first equation:

xleft(frac{9}{2} - xright) 5

Expanding this equation:

frac{9}{2}x - x^2 5

Rearranging the equation to get a standard quadratic form:

-x^2 frac{9}{2}x - 5 0

To eliminate the fraction, multiply the entire equation by 2:

-2x^2 9x - 10 0

Multiplying through by -1:

2x^2 - 9x 10 0

Now, we can use the quadratic formula to solve for x:

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

Here, a 2, b -9, and c 10:

x frac{9 pm sqrt{(-9)^2 - 4 cdot 2 cdot 10}}{2 cdot 2}

Calculating the discriminant:

(-9)^2 - 4 cdot 2 cdot 10 81 - 80 1

Substituting back into the formula:

x frac{9 pm 1}{4}

This gives us two possible values for x:

x frac{10}{4} frac{5}{2}
and
x frac{8}{4} 2

Now, substituting these back to find y:

For x 5/2: y frac{9}{2} - frac{5}{2} 2 For x 2: y frac{9}{2} - 2 frac{5}{2}

Thus, the two numbers are 5/2 and 2.

Algebraic Solution by Yuusuf Abdulahi Mahamed

Yuusuf Abdulahi Mahamed provides an alternative algebraic solution.

Let x and y be the two numbers we are going to find. The product of the two numbers is 5: xy 5 The sum of the two numbers is 9/2: x y frac{9}{2}

From the first equation, we can express y in terms of x:

y frac{5}{x}

Substitute this in the second equation:

x cdot frac{5}{x} frac{9}{2}

Multiplying through by x:

5 frac{9x}{2}

Multiplying through by 2:

10 9x

Thus:

x frac{10}{9}

Substituting this back into the expression for y:

y frac{5}{left(frac{10}{9}right)} frac{5 cdot 9}{10} frac{9}{2}

However, we need to check the factorization approach for verification:

2x2 - 9x 10 0

Factoring the quadratic equation:

(2x - 5)(x - 2) 0

Two possibilities:

2x - 5 0
thus: x frac{5}{2}
and thus: y frac{9}{2} - frac{5}{2} 2 x - 2 0
thus: x 2
and thus: y frac{9}{2} - 2 frac{5}{2}

Thus, the two numbers are 5/2 and 2.

Thank you for the challenge, Yuusuf. Enjoy your day.

Conclusion

Both solutions confirm that the two numbers are 5/2 and 2, satisfying the conditions that their product is 5 and their sum is 9/2. Using algebraic methods, we successfully solved this problem, showcasing the power of quadratic equations and algebraic manipulation in solving real-world problems.