the sum of two numbers is 42. their product is 185. what are the two numbers?
37 and 5
x+y = 42 ---> y = 42-x
xy = 185
x(42-x) = 185
42x - x^2 - 185 = 0
x^2 - 42x + 185 = 0
x^2 - 42x +441 = -185+441
(x-21)^2 = 256
x-21 = ± √256
x = 21 ±16 = 37 or 5
if one number is 37 , the other is 5
or one is 5 , the other is 37
check:
sum = 37+5 = 42
product = 37x5 = 185
9b+21
To find the two numbers, let's consider algebraic approach:
Let's assume the two numbers as x and y.
According to the given information, we have two equations:
1. x + y = 42
2. x * y = 185
To solve this system of equations, we can use substitution or elimination method. In this case, let's solve it using the substitution method.
Step 1: Solve the first equation for either x or y in terms of the other variable.
For example, let's solve equation 1 for x:
x = 42 - y
Step 2: Substitute the expression we found in step 1 into the second equation.
Replacing x in equation 2, we get:
(42 - y) * y = 185
Step 3: Simplify and solve the resulting quadratic equation.
Expanding the equation, we have:
42y - y^2 = 185
Rearranging the equation, we get:
y^2 - 42y + 185 = 0
Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula.
Step 4: Solve the quadratic equation to find the values of y.
Using the quadratic formula, we get:
y = (42 ± √(42^2 - 4 * 1 * 185)) / (2 * 1)
Simplifying further, we have:
y = (42 ± √(1764 - 740)) / 2
y = (42 ± √1024) / 2
y = (42 ± 32) / 2
Solving for y, we have two possible solutions:
y = (42 + 32) / 2 = 74 / 2 = 37
y = (42 - 32) / 2 = 10 / 2 = 5
Step 5: Substitute the found values of y back into equation 1 to find the corresponding values of x.
Using the value y = 37 in equation 1, we find:
x + 37 = 42
x = 42 - 37
x = 5
Using the value y = 5 in equation 1, we find:
x + 5 = 42
x = 42 - 5
x = 37
So, the two numbers are 5 and 37, or vice versa.