The sum of two numbers is 30. Their product is 176. What are two numbers?
To find the two numbers, let's assign variables to them. Let's call the first number x and the second number y.
According to the given information, we have two equations:
Equation 1: x + y = 30 (the sum of the two numbers is 30)
Equation 2: xy = 176 (the product of the two numbers is 176)
To solve this system of equations, we can use substitution or elimination method. Let's use substitution method in this case.
From Equation 1, we can isolate one variable. Let's solve it for x:
x = 30 - y
Now we substitute this value of x into Equation 2:
(30 - y) * y = 176
Simplifying the equation by multiplying:
30y - y^2 = 176
Rearranging the equation in standard quadratic form:
y^2 - 30y + 176 = 0
Now we can solve this quadratic equation using factoring, completing the square, or using the quadratic formula.
Factoring doesn't work in this case, so let's use the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 1, b = -30, and c = 176. Substituting the values into the formula:
y = (-(-30) ± √((-30)^2 - 4 * 1 * 176)) / (2 * 1)
Simplifying:
y = (30 ± √(900 - 704)) / 2
y = (30 ± √196) / 2
y = (30 ± 14) / 2
Using both possibilities for y, we get two potential values:
1. y = (30 + 14) / 2 = 44 / 2 = 22
2. y = (30 - 14) / 2 = 16 / 2 = 8
Now that we have the values for y, we can substitute them back into Equation 1 to find the corresponding values for x.
For y = 22:
x = 30 - y = 30 - 22 = 8
So the first number is 8 and the second number is 22.
For y = 8:
x = 30 - y = 30 - 8 = 22
So the first number is 22 and the second number is 8.
Thus, the two numbers are 8 and 22.
x+y = 30
xy = 176
from the firs: y = 30-x
x(30-x) = 176
30x - x^2 - 176 = 0
x^2 - 30x + 176 = 0
hint: it factors, if you can't find the factors, use the quadratic formula.