The product of two numbers is 120. The sum of the two numbers is 22. What are the two numbers?
Question
The product of two numbers is 120.
The sum of the two numbers is 22.
What are the two numbers?
Solution
Let's denote the two numbers as x and y.
From the problem, we have two equations:
- x*y = 120 (The product of the two numbers is 120)
- x + y = 22 (The sum of the two numbers is 22)
We can solve this system of equations step by step:
Step 1: Solve the second equation for one variable. Let's solve for y:
y = 22 - x
Step 2: Substitute y in the first equation:
x * (22 - x) = 120 22x - x^2 = 120 Rearrange the equation: x^2 - 22x + 120 = 0
Step 3: Solve the quadratic equation. This can be done by factoring, completing the square, or using the quadratic formula. The factored form of the equation is:
(x - 10)(x - 12) = 0
Step 4: Set each factor equal to zero and solve for x:
x - 10 = 0 or x - 12 = 0
So, x = 10 or x = 12
Step 5: Substitute x = 10 and x = 12 into the equation y = 22 - x to find the corresponding y values:
If x = 10, then y = 22 - 10 = 12 If x = 12, then y = 22 - 12 = 10
So, the two numbers are 10 and 12.
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