What two numbers multiply to 189 but add to 66?
The Correct Answer and Explanation is:
To find two numbers that multiply to 189 and add to 66, we can denote these two numbers as ( x ) and ( y ). The problem gives us two equations based on the relationships between ( x ) and ( y ):
- ( x \cdot y = 189 ) (the product)
- ( x + y = 66 ) (the sum)
We can start by expressing one variable in terms of the other using the sum equation. Let’s solve for ( y ):
[
y = 66 – x
]
Next, we can substitute this expression for ( y ) into the product equation:
[
x \cdot (66 – x) = 189
]
Expanding this equation gives us:
[
66x – x^2 = 189
]
Rearranging this into a standard quadratic equation form, we have:
[
-x^2 + 66x – 189 = 0
]
To make it easier to work with, we can multiply the entire equation by -1:
[
x^2 – 66x + 189 = 0
]
Now, we can solve this quadratic equation using the quadratic formula, which is:
[
x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}
]
In our equation, ( a = 1 ), ( b = -66 ), and ( c = 189 ). Plugging these values into the quadratic formula:
[
x = \frac{66 \pm \sqrt{(-66)^2 – 4 \cdot 1 \cdot 189}}{2 \cdot 1}
]
Calculating the discriminant:
[
b^2 – 4ac = 4356 – 756 = 3600
]
Taking the square root gives us:
[
\sqrt{3600} = 60
]
Now we can substitute this back into the formula:
[
x = \frac{66 \pm 60}{2}
]
Calculating the two possible values for ( x ):
- ( x = \frac{126}{2} = 63 )
- ( x = \frac{6}{2} = 3 )
Thus, the two numbers are ( 63 ) and ( 3 ).
We can verify:
- Product: ( 63 \times 3 = 189 )
- Sum: ( 63 + 3 = 66 )
Therefore, the two numbers that multiply to 189 and add to 66 are 63 and 3.