To solve this problem, we need to define the two numbers and create equations based on the given information. Let's call the two numbers [tex]\( a \)[/tex] and [tex]\( b \)[/tex], where [tex]\( a \)[/tex] is the larger number.
Given:
1. The product of two numbers is 572.
[tex]\[ a \cdot b = 572 \][/tex]
2. The sum of the numbers is 12 times their positive difference.
[tex]\[ a + b = 12 \cdot |a - b| \][/tex]
We also know that since [tex]\( a \)[/tex] is the larger number, the absolute value is not necessary since [tex]\( a - b \)[/tex] will always be positive. Therefore, we can rewrite the second equation as:
[tex]\[ a + b = 12(a - b) \][/tex]
Now let's solve the equations.
Equation 1:
[tex]\[ a \cdot b = 572 \][/tex]
[tex]\[ b = \frac{572}{a} \][/tex]
Equation 2:
[tex]\[ a + b = 12(a - b) \][/tex]
[tex]\[ a + \frac{572}{a} = 12a - 12\frac{572}{a} \][/tex]
To eliminate the fraction, multiply through by [tex]\( a \)[/tex]:
[tex]\[ a^2 + 572 = 12a^2 - 6876 \][/tex]
Reorder terms and combine like terms:
[tex]\[ 11a^2 - a^2 = 6876 + 572 \][/tex]
[tex]\[ 11a^2 - a^2 = 7448 \][/tex]
[tex]\[ 10a^2 = 7448 \][/tex]
[tex]\[ a^2 = \frac{7448}{10} \][/tex]
[tex]\[ a^2 = 744.8 \][/tex]
Since [tex]\( a \)[/tex] is a real number, we take the square root of both sides:
[tex]\[ a = \pm\sqrt{744.8} \][/tex]
However, we are looking for the larger number which must be positive:
[tex]\[ a = \sqrt{744.8} \][/tex]
The square root is slightly below [tex]\( \sqrt{729} = 27 \)[/tex] and we are looking for integers since the product of the numbers is an integer and the sum also suggests they should be integers. Therefore, the closest integer is [tex]\( a = 26 \)[/tex]. But we must verify if [tex]\( b \)[/tex], the smaller number, is also an integer.
[tex]\[ b = \frac{572}{a} \][/tex]
[tex]\[ b = \frac{572}{26} \][/tex]
[tex]\[ b = 22 \][/tex]
So the two numbers that satisfy both conditions are 26 and 22. Since [tex]\( a \)[/tex] is the larger number:
The larger of the two numbers is 26.
