Asked by Ur m0m
The sum of two numbers is 22, and the product of the same two numbers is 120. What are the two numbers? Input the lesser number first. (1 point)
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Answered by
GPT-4o mini
Let the two numbers be \( x \) and \( y \). According to the problem, we have the following two equations:
1. \( x + y = 22 \)
2. \( x \cdot y = 120 \)
From the first equation, we can express \( y \) in terms of \( x \):
\[ y = 22 - x \]
Now, we can substitute \( y \) in the second equation:
\[ x(22 - x) = 120 \]
Expanding this, we get:
\[ 22x - x^2 = 120 \]
Rearranging the equation results in:
\[ x^2 - 22x + 120 = 0 \]
Next, we can factor this quadratic equation. We need two numbers that multiply to 120 and add up to -22. These numbers are -12 and -10:
\[ (x - 12)(x - 10) = 0 \]
Setting each factor to zero gives us the solutions:
\[ x - 12 = 0 \quad \Rightarrow \quad x = 12 \]
\[ x - 10 = 0 \quad \Rightarrow \quad x = 10 \]
Thus, the two numbers are 10 and 12. Since we need to input the lesser number first, the answer is:
**10, 12**
1. \( x + y = 22 \)
2. \( x \cdot y = 120 \)
From the first equation, we can express \( y \) in terms of \( x \):
\[ y = 22 - x \]
Now, we can substitute \( y \) in the second equation:
\[ x(22 - x) = 120 \]
Expanding this, we get:
\[ 22x - x^2 = 120 \]
Rearranging the equation results in:
\[ x^2 - 22x + 120 = 0 \]
Next, we can factor this quadratic equation. We need two numbers that multiply to 120 and add up to -22. These numbers are -12 and -10:
\[ (x - 12)(x - 10) = 0 \]
Setting each factor to zero gives us the solutions:
\[ x - 12 = 0 \quad \Rightarrow \quad x = 12 \]
\[ x - 10 = 0 \quad \Rightarrow \quad x = 10 \]
Thus, the two numbers are 10 and 12. Since we need to input the lesser number first, the answer is:
**10, 12**
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