What are two numbers that multiply to 24 AND add to 8?
The Correct Answer and Explanation is:
To find two numbers that multiply to 24 and add to 8, we can define the two numbers as ( x ) and ( y ). We need to satisfy the following two equations:
- ( x \cdot y = 24 )
- ( x + y = 8 )
To solve this system of equations, we can express one variable in terms of the other using the second equation. From ( x + y = 8 ), we can solve for ( y ):
[
y = 8 – x
]
Next, we can substitute this expression for ( y ) into the first equation:
[
x \cdot (8 – x) = 24
]
Expanding this equation gives:
[
8x – x^2 = 24
]
Rearranging this to set it to zero yields:
[
x^2 – 8x + 24 = 0
]
Next, we can apply the quadratic formula to find the values of ( x ):
[
x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}
]
In this case, ( a = 1 ), ( b = -8 ), and ( c = 24 ). Plugging in these values:
[
x = \frac{8 \pm \sqrt{(-8)^2 – 4 \cdot 1 \cdot 24}}{2 \cdot 1}
]
[
x = \frac{8 \pm \sqrt{64 – 96}}{2}
]
[
x = \frac{8 \pm \sqrt{-32}}{2}
]
Since the term under the square root is negative, ( \sqrt{-32} ) indicates that there are no real solutions to this equation. Therefore, there are no two real numbers that both multiply to 24 and add to 8.
However, we can explore the problem with integer pairs that multiply to 24. The integer pairs that multiply to 24 are (1, 24), (2, 12), (3, 8), (4, 6). Among these pairs, none add up to 8.
In conclusion, there are no two numbers that satisfy both conditions of multiplying to 24 and adding to 8 within the set of real numbers. The mathematical operations and reasoning demonstrate the inability to find such pairs, emphasizing the importance of checking both conditions systematically.