The product of two integers is -24. The sum of the two integers is -2. What are the numbers?
-6 + 4 = -2
or, if you want to do the math,
ab = -24
a+b = -2
so,
a - 24/a = -2
a^2 + 2a - 24 = 0
(a+6)(a-4) = 0
idk
To solve this problem, we can use a system of equations. Let's call the two integers x and y.
We are given two pieces of information:
1. The product of the two integers is -24. This can be written as xy = -24.
2. The sum of the two integers is -2. This can be written as x + y = -2.
Now we can solve the system of equations.
We can start by rearranging the first equation to solve for one variable in terms of the other:
x = -24/y.
Substituting this value of x into the second equation, we get:
(-24/y) + y = -2.
To simplify the equation, we can multiply both sides by y to get rid of the fraction:
-24 + y^2 = -2y.
Rearranging this equation, we have:
y^2 + 2y - 24 = 0.
Now we can factor this quadratic equation:
(y + 6)(y - 4) = 0.
Setting each factor equal to zero, we have:
y + 6 = 0 --> y = -6,
and
y - 4 = 0 --> y = 4.
So the two possible values for y are -6 and 4.
Now we can substitute these values back into the first equation to solve for x.
For y = -6:
x = -24/(-6) = 4.
For y = 4:
x = -24/4 = -6.
Therefore, the two numbers are 4 and -6, or -6 and 4.
To find the two integers, let's assign variables to represent them. Let's call the first integer x and the second integer y.
We can translate the given information into equations:
Equation 1: x * y = -24 (The product of two integers is -24)
Equation 2: x + y = -2 (The sum of the two integers is -2)
Now we have a system of equations. We can solve it using various methods, but let's use substitution.
Solving Equation 2 for one variable:
x + y = -2
y = -2 - x (subtracting x from both sides)
Substitute the value of y in Equation 1:
x * (-2 - x) = -24
Simplify and solve the equation:
-2x - x² = -24
Rearrange the equation:
x² + 2x - 24 = 0
Now we have a quadratic equation. We can factor it or use the quadratic formula to solve for x. Let's factor it:
(x + 6)(x - 4) = 0
By setting each factor equal to zero, we get two possible solutions for x:
x + 6 = 0 --> x = -6
x - 4 = 0 --> x = 4
Now that we have the values of x, we can substitute them back into Equation 2 to find y:
For x = -6:
(-6) + y = -2
y = -2 + 6
y = 4
For x = 4:
4 + y = -2
y = -2 - 4
y = -6
Therefore, the two numbers that satisfy the given conditions are -6 and 4.