A number is 8 more than another number. The product of these two numbers is 20. Find the numbers.
let:
n=the number
m=the other number
n=8+m-----eqtn 1
nXm=20-----eqtn.2
substitute eqtn1 to eqtn2
(8+m)(m)=20
8m+m^2=20
arrange
m^2+8m-20=0
it forms into quadratic eqtn.
factor by grouping
(m-2)(m+10)=0
by zero property of equality we have:
m=2 and m=-10
-10 is an extraneous root since it cannot satisfy the equation.
there should'nt be negative
use m=2
substitute to eqtn.1
n=8+2
n=10
therefore the numbers are: 2 and 10
check:
the product of the numbers is 20:
2X10 = 20 check!
the number is 8 more than the other number:
n=8+2= 10 check!
Don't rule out m = -10 as extraneous, since they do satisfy the given conditionn.
of m = -10, m+8 = -2
their product = (-10)(-1) = 20
-2 is 8 more than -10
the numbers could also be -2 and -10
For the following two numbers, find two factors of the first number such that their product is the first number and their sum is the second number.
44,15
To solve this problem, let's assign variables to the two unknown numbers. Let's call one of the numbers "x" and the other number "y".
According to the given information, we can set up two equations:
1) "A number is 8 more than another number": This can be written as x = y + 8.
2) "The product of these two numbers is 20": This can be written as xy = 20.
Now, we have a system of equations to solve:
Equation 1: x = y + 8
Equation 2: xy = 20
To solve this system, we can substitute the value of x from Equation 1 into Equation 2:
(y + 8)y = 20
Expanding this equation, we get:
y^2 + 8y = 20
Rearranging the equation, we have:
y^2 + 8y - 20 = 0
Now, we can solve this quadratic equation by factoring or using the quadratic formula. Let's use the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / 2a
For this equation, a = 1, b = 8, and c = -20:
y = (-8 ± √(8^2 - 4(1)(-20))) / (2 * 1)
Calculating the values inside the square root, we have:
y = (-8 ± √(64 + 80)) / 2
y = (-8 ± √144) / 2
Simplifying, we get:
y = (-8 ± 12) / 2
Now, we consider both possibilities:
1) When y = (-8 + 12) / 2 = 4 / 2 = 2
2) When y = (-8 - 12) / 2 = -20 / 2 = -10
Now that we have the values of y, we can substitute them back into Equation 1 to find the corresponding values of x:
For y = 2:
x = 2 + 8 = 10
For y = -10:
x = -10 + 8 = -2
Therefore, the two numbers are 10 and 2, or -2 and -10.