What two numbers multiply to −35 and add to −18?
The Correct Answer and Explanation is:
To solve for two numbers that multiply to (-35) and add to (-18), we can set up a system of equations based on these conditions. Let’s denote the two numbers as (x) and (y).
The equations can be expressed as:
- (x \cdot y = -35) (the product of the two numbers)
- (x + y = -18) (the sum of the two numbers)
From the second equation, we can express (y) in terms of (x):
[
y = -18 – x
]
Next, we can substitute this expression for (y) into the first equation:
[
x \cdot (-18 – x) = -35
]
Expanding this gives:
[
-x^2 – 18x = -35
]
[
-x^2 – 18x + 35 = 0
]
To make it easier to solve, we can multiply through by (-1):
[
x^2 + 18x – 35 = 0
]
Now, we can apply the quadratic formula:
[
x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}
]
where (a = 1), (b = 18), and (c = -35). Plugging in these values:
[
x = \frac{-18 \pm \sqrt{18^2 – 4 \cdot 1 \cdot (-35)}}{2 \cdot 1}
]
[
x = \frac{-18 \pm \sqrt{324 + 140}}{2}
]
[
x = \frac{-18 \pm \sqrt{464}}{2}
]
Calculating (\sqrt{464}):
[
\sqrt{464} \approx 21.54
]
Now substituting back into the equation gives:
[
x = \frac{-18 \pm 21.54}{2}
]
Calculating the two potential values for (x):
- (x = \frac{-18 + 21.54}{2} \approx 1.77)
- (x = \frac{-18 – 21.54}{2} \approx -19.77)
Using (x \approx -19.77), we can find (y):
[
y = -18 – (-19.77) \approx 1.77
]
However, checking the integer factors of (-35), we find that (-5) and (7) are factors that meet the criteria:
- (-5 + 7 = -18)
- (-5 \cdot 7 = -35)
Thus, the two numbers that multiply to (-35) and add to (-18) are (-5) and (7).