What Are Two Numbers Whose Sum Is -4 and Their Product Is -3?
The problem involves finding two numbers where the sum is -4 and their product is -3. This can be approached using algebraic methods, including solving a system of equations and understanding quadratic equations.
System of Equations
Let's denote the two unknown numbers by (x) and (y).
We have the following two equations:
1. (x y -4)
2. (xy -3)
We can express one variable in terms of the other from the first equation:
(y -4 - x)
Substitute (y) in the second equation:
(x(-4 - x) -3)
Simplify to get a quadratic equation:
(-4x - x^2 3 0)
(x^2 4x - 3 0)
Now, we can solve this quadratic equation using the quadratic formula:
(x frac{-b pm sqrt{b^2 - 4ac}}{2a})
Here, (a 1), (b 4), and (c -3).
(x frac{-4 pm sqrt{4^2 - 4 cdot 1 cdot (-3)}}{2 cdot 1})
(x frac{-4 pm sqrt{16 12}}{2})
(x frac{-4 pm sqrt{28}}{2})
(x frac{-4 pm 2sqrt{7}}{2})
(x -2 pm sqrt{7})
Thus, the solutions for (x) are:
(x -2 sqrt{7})
(x -2 - sqrt{7})
For each (x), we can find (y) using (y -4 - x).
When (x -2 sqrt{7}):
(y -4 - (-2 sqrt{7}))
(y -2 - sqrt{7})
When (x -2 - sqrt{7}):
(y -4 - (-2 - sqrt{7}))
(y -2 sqrt{7})
Therefore, the two pairs of numbers are:
((-2 sqrt{7}, -2 - sqrt{7}))
((-2 - sqrt{7}, -2 sqrt{7}))
Alternatively, Using a Different Approach
Consider the given conditions:
1. (xy -4)
2. (xy -3)
This seems contradictory, as (xy) cannot be equal to two different values simultaneously. Therefore, there might be an error or a misunderstanding in the problem statement.
Let's correct the problem: Assume the correct product is -3, and solve it as follows:
1. (xy -3)
2. (xy -3)
Let (x -3/y):
(-3/y cdot y -3)
This simplifies to:
(-3 -3)
Now, we need (x y -4):
(-3/y y -4)
Multiply both sides by (y):
(-3 y^2 -4y)
Rearrange to form a quadratic equation:
(y^2 4y - 3 0)
Use the quadratic formula to solve for (y):
(y frac{-4 pm sqrt{4^2 - 4 cdot 1 cdot (-3)}}{2 cdot 1})
(y frac{-4 pm sqrt{16 12}}{2})
(y frac{-4 pm sqrt{28}}{2})
(y frac{-4 pm 2sqrt{7}}{2})
(y -2 pm sqrt{7})
Thus, the solutions for (y) are:
(y -2 sqrt{7})
(y -2 - sqrt{7})
For each (y), we can find (x):
When (y -2 sqrt{7}):
(x -3/(-2 sqrt{7}))
Rationalize the denominator:
(x frac{-3}{-2 sqrt{7}} cdot frac{-2 - sqrt{7}}{-2 - sqrt{7}} frac{-3(-2 - sqrt{7})}{4 - 7} frac{6 3sqrt{7}}{-3} -2 - sqrt{7})
When (y -2 - sqrt{7}):
(x -3/(-2 - sqrt{7}))
Rationalize the denominator:
(x frac{-3}{-2 - sqrt{7}} cdot frac{-2 sqrt{7}}{-2 sqrt{7}} frac{-3(-2 sqrt{7})}{4 - 7} frac{6 - 3sqrt{7}}{-3} -2 sqrt{7})
Therefore, the pairs of numbers are:
((-2 - sqrt{7}, -2 sqrt{7}))
((-2 sqrt{7}, -2 - sqrt{7}))
Conclusion
The numbers (x) and (y) that satisfy the given conditions are ((-2 sqrt{7}, -2 - sqrt{7})) and ((-2 - sqrt{7}, -2 sqrt{7})).
Above methods and steps are straightforward and utilize algebraic techniques to solve the problem effectively.