Finding the Numbers Whose Sum and Product Are Given

Consider the scenario where the sum of two numbers is 12 and their product is 32. This problem can be solved using basic algebraic techniques, particularly through the use of quadratic equations. This article will guide you through the steps to find these numbers and discuss the broader application of such problems in mathematics.

Understanding the Problem

The problem statement is straightforward: we are given that the sum of two numbers is 12, i.e., ( x y 12 ), and the product of these numbers is 32, i.e., ( xy 32 ).

Solving the Problem Using Quadratic Equations

To find the values of ( x ) and ( y ), we will use these two equations and solve for one of the variables.

Step 1: Expressing One Variable in Terms of the Other

We start by expressing ( y ) in terms of ( x ).

Equation 1: ( x y 12 )

Solving Equation 1 for ( y ):
( y 12 - x )

Step 2: Substituting the Expression into the Second Equation

Next, we substitute ( y 12 - x ) into Equation 2, ( xy 32 ):
( x(12 - x) 32 )

Step 3: Expanding and Rearranging

Expanding and rearranging the equation:

( 12x - x^2 32 )

Bringing all terms to one side, we get a quadratic equation:

( x^2 - 12x 32 0 )

Step 4: Solving the Quadratic Equation

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

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

Here, ( a 1 ), ( b -12 ), and ( c 32 ). Plugging these values into the formula:

( x frac{12 pm sqrt{(-12)^2 - 4 cdot 1 cdot 32}}{2 cdot 1} )

Calculating the discriminant:

( (-12)^2 - 4 cdot 1 cdot 32 144 - 128 16 )

Solving for ( x ):

( x frac{12 pm 4}{2} )

( x 8 ) or ( x 4 )

Step 5: Identifying Two Possible Values for ( x ) and ( y )

When ( x 8 ), ( y 12 - 8 4 ).

When ( x 4 ), ( y 12 - 4 8 ).

Thus, the two numbers are ( 8 ) and ( 4 ).

Step 6: Verifying the Solution

Check the sum and product conditions:

Sum: ( 8 4 12 ) (True)

Product: ( 8 times 4 32 ) (True)

The solution satisfies both conditions.

Conclusion

The difference between the two numbers is ( 8 - 4 4 ).

Broader Application of These Concepts

This type of problem is not only useful in algebra but also in various fields like computer science, physics, and engineering. Understanding these concepts helps in solving real-world problems like optimizing functions, understanding relationships between variables, and more.