the sum of two numbers is 42. their product is 185. what are the two numbers
let the numbers be x and y
x+y = 42 ---> y = 42-x
xy = 185 --->x(42-x) = 185
42x - x^2 - 185 = 0
x^2 - 42x + 185 = 0
(x-37)(x - 5) 0
x = 37 or x = 5
then y = 42-37 or y = 42-5
y = 5 or y = 37
(symmetric solution)
the two numbers are 5 and 37
To find the two numbers, let's call them x and y. We are given two pieces of information:
1. The sum of the two numbers is 42, which can be written as:
x + y = 42
2. The product of the two numbers is 185, which can be written as:
xy = 185
To solve this system of equations, we can use substitution or elimination method. Let's solve it using the substitution method:
From equation 1, we can rearrange it to get:
x = 42 - y
Now we substitute this value of x into equation 2:
(42 - y)y = 185
Expanding the equation:
42y - y^2 = 185
Rearranging the equation to bring it to quadratic form:
y^2 - 42y + 185 = 0
Now we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. In this case, the equation doesn't factor easily, so let's use the quadratic formula:
The quadratic formula is given by:
y = (-b ± √(b^2 - 4ac)) / 2a
For our quadratic equation in the form of ay^2 + by + c = 0, substituting the values:
a = 1, b = -42, c = 185
y = (-(-42) ± √((-42)^2 - 4(1)(185))) / 2(1)
Simplifying:
y = (42 ± √(1764 - 740)) / 2
y = (42 ± √(1024)) / 2
y = (42 ± 32) / 2
Now we have two possible values for y. Let's calculate both:
y1 = (42 + 32) / 2
= 74 / 2
= 37
y2 = (42 - 32) / 2
= 10 / 2
= 5
Now, we substitute these values of y into equation 1 to find x:
For y1 = 37:
x + 37 = 42
x = 42 - 37
x = 5
For y2 = 5:
x + 5 = 42
x = 42 - 5
x = 37
So, the two numbers are 5 and 37.
To find the two numbers, we can set up a system of equations based on the given information.
Let's say the two numbers are x and y.
According to the problem, the sum of the two numbers is 42, so we can write the equation:
x + y = 42 (equation 1)
Also, it states that their product is 185, so we can write another equation:
x * y = 185 (equation 2)
Now we have a system of equations with two unknowns. To solve this system, we can use a method called substitution or elimination:
Method 1 - Substitution:
1. Solve equation 1 for x:
x = 42 - y
2. Substitute x in equation 2 with the value from step 1:
(42 - y) * y = 185
Simplify the equation:
42y - y^2 = 185
3. Rearrange the equation to put it in quadratic form:
y^2 - 42y + 185 = 0
Now, we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. Once we find the value of y, we can substitute it back into equation 1 to find the corresponding value of x.
Method 2 - Elimination:
1. Multiply equation 1 by -1:
-x - y = -42
2. Add equation 2 and the modified equation 1:
(x * y) + (-x - y) = 185 - 42
Simplify the equation:
-y = 143
3. Solve for y:
y = -143
Substitute the value of y into equation 1 to find x:
x + (-143) = 42
Simplify the equation:
x - 143 = 42
Add 143 to both sides:
x = 42 + 143
Simplify the expression:
x = 185
Therefore, the two numbers are x = 185 and y = -143.