What times what equals 12 but when you add those numbers together it equals 3
The Correct Answer and Explanation is :
The Correct Answer is : ( 5.275 ) and ( -2.275 )
To solve the equation where two numbers multiply to 12 and add up to 3, we can set up the equations based on the given conditions.
Let’s denote the two numbers as ( x ) and ( y ). According to the problem, we have:
- ( x \cdot y = 12 )
- ( x + y = 3 )
We can solve these equations step by step.
Step 1: Express One Variable in Terms of the Other
From the second equation ( x + y = 3 ), we can express ( y ) in terms of ( x ):
[
y = 3 – x
]
Step 2: Substitute into the First Equation
Next, we can substitute this expression for ( y ) into the first equation:
[
x \cdot (3 – x) = 12
]
Step 3: Expand and Rearrange the Equation
Expanding the left side gives:
[
3x – x^2 = 12
]
Rearranging this equation leads to:
[
-x^2 + 3x – 12 = 0
]
Multiplying through by -1 to make the leading coefficient positive:
[
x^2 – 3x + 12 = 0
]
Step 4: Use the Quadratic Formula
Now, we can apply the quadratic formula, where ( a = 1 ), ( b = -3 ), and ( c = -12 ):
[
x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}
]
Substituting in the values:
[
x = \frac{3 \pm \sqrt{(-3)^2 – 4 \cdot 1 \cdot (-12)}}{2 \cdot 1}
]
This simplifies to:
[
x = \frac{3 \pm \sqrt{9 + 48}}{2}
]
[
x = \frac{3 \pm \sqrt{57}}{2}
]
Step 5: Finding the Values of ( x ) and ( y )
Calculating ( \sqrt{57} \approx 7.55 ):
- ( x = \frac{3 + 7.55}{2} \approx 5.275 )
- ( x = \frac{3 – 7.55}{2} \approx -2.275 )
Substituting back to find ( y ):
- For ( x \approx 5.275 ), ( y \approx 3 – 5.275 \approx -2.275 ).
- For ( x \approx -2.275 ), ( y \approx 3 – (-2.275) \approx 5.275 ).
Conclusion
The two numbers that multiply to 12 and add up to 3 are approximately ( 5.275 ) and ( -2.275 ). This illustrates how the properties of numbers can lead to non-integer solutions, showcasing the relationships between multiplication and addition in algebra.