The sum of two numbers is 4 and their product is -32
Find the numbers.
-4 and + 8
x+y = 4
xy = -32
so from the first: y = 4-x
into the second
x(4-x) = -32
4x - x^2 + 32 = 0
x^2 - 4x - 32 = 0
(x-8)(x+4) - 0
x = 8 or x = -4
if x=8 then y = -4
if x = -4 then y = 8
the numbers are 8 and -4
To find the numbers, let's assume the two numbers are x and y.
1. The sum of the two numbers is 4, so we can write the equation:
x + y = 4
2. The product of the two numbers is -32, so we can write the equation:
xy = -32
To solve this system of equations, we can use substitution or elimination method. Let's use the substitution method.
From equation 1, we can rewrite it as:
y = 4 - x
Substitute this value of y into equation 2:
x(4 - x) = -32
Expand the equation:
4x - x^2 = -32
Rearrange the equation:
x^2 - 4x - 32 = 0
Factor the equation:
(x - 8)(x + 4) = 0
Setting each factor equal to zero and solving for x:
x - 8 = 0 or x + 4 = 0
Solving for x:
x = 8 or x = -4
Now substitute the value of x back into equation 1 to find y:
For x = 8:
8 + y = 4
y = 4 - 8
y = -4
For x = -4:
-4 + y = 4
y = 4 + 4
y = 8
Therefore, the numbers are:
x = 8, y = -4
or
x = -4, y = 8
To find the two numbers, let's use a system of equations.
Let's call the two numbers x and y.
According to the problem, we know that the sum of the two numbers is 4:
x + y = 4 -- Equation 1
We also know that their product is -32:
xy = -32 -- Equation 2
To solve this system of equations, we can use the method of substitution or elimination.
Let's solve by substitution. From Equation 1, we can solve for one variable in terms of the other. Let's solve for y:
y = 4 - x
Now we can substitute this expression for y into Equation 2:
x(4 - x) = -32
Expanding the equation:
4x - x^2 = -32
Rearranging this quadratic equation:
x^2 - 4x - 32 = 0
Now we need to factor or use the quadratic formula to find the values of x. Factoring may not be straightforward in this case, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
For this equation, a = 1, b = -4, and c = -32. Substituting these values into the quadratic formula:
x = (-(-4) ± √((-4)^2 - 4*1*(-32))) / (2*1)
x = (4 ± √(16 + 128)) / 2
x = (4 ± √144) / 2
x = (4 ± 12) / 2
Simplifying:
x = (4 + 12) / 2 or x = (4 - 12) / 2
x = 16 / 2 or x = -8 / 2
x = 8 or x = -4
Now that we know the possible values for x, we can substitute them back into Equation 1 to find the corresponding values for y:
For x = 8:
y = 4 - x
y = 4 - 8
y = -4
For x = -4:
y = 4 - x
y = 4 - (-4)
y = 8
Therefore, the two numbers that satisfy both equations are 8 and -4.