In the realm of mathematics, there are often intriguing questions that challenge our understanding of numbers. One such question is: What two numbers multiply to 6 but add up to 2? This problem encourages us to explore relationships between multiplication and addition in a way that might not be immediately obvious.
Breaking Down the Problem
Let's denote the two numbers we are looking for as x and y. According to the problem, we have the following two equations:
-
Multiplication:
( x \times y = 6 ) -
Addition:
( x + y = 2 )
Solving the Equations
To find x and y, we can use the second equation to express y in terms of x:
[ y = 2 - x ]
Now, we can substitute this expression for y into the first equation:
[ x \times (2 - x) = 6 ]
Expanding this gives us:
[ 2x - x^2 = 6 ]
Rearranging the equation leads to:
[ -x^2 + 2x - 6 = 0 ]
Multiplying through by -1 simplifies this to:
[ x^2 - 2x + 6 = 0 ]
Finding the Roots
To find the roots of the quadratic equation, we apply the quadratic formula:
[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]
Here, ( a = 1 ), ( b = -2 ), and ( c = 6 ).
Calculating the discriminant ( b^2 - 4ac ):
[ (-2)^2 - 4 \times 1 \times 6 = 4 - 24 = -20 ]
Since the discriminant is negative, it indicates that there are no real solutions for x and y that satisfy both conditions simultaneously. Instead, we will find complex solutions.
Complex Solutions
Now we can compute the complex solutions:
[ x = \frac{2 \pm \sqrt{-20}}{2} ]
This can be rewritten as:
[ x = \frac{2 \pm 2i\sqrt{5}}{2} ]
Simplifying gives:
[ x = 1 \pm i\sqrt{5} ]
Substituting back to find y:
[ y = 2 - (1 \pm i\sqrt{5}) = 1 \mp i\sqrt{5} ]
Thus, the solutions are:
- x = 1 + i√5, y = 1 - i√5
- or vice versa.
Conclusion
In conclusion, while it may be intriguing to think about pairs of numbers that multiply to 6 and add up to 2, the reality is that such pairs do not exist among real numbers. Instead, we find that the numbers involved are complex, highlighting the fascinating interplay between different branches of mathematics. Understanding these relationships not only enhances our problem-solving skills but also deepens our appreciation for the intricacies of numbers.
