Two consecutive positive integers have the property that one integer times twice the other equals 612. What is the sum of these two integers?
The integers are 17 and 18 (17 x 36 = 612, so the answer is 35.
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Let's assume the two consecutive positive integers are x and x + 1.
According to the given statement, one integer times twice the other equals 612:
x * (2(x + 1)) = 612
Expanding the equation:
2x(x + 1) = 612
Multiplying:
2x^2 + 2x = 612
Moving all terms to one side:
2x^2 + 2x - 612 = 0
Dividing the entire equation by 2 to simplify:
x^2 + x - 306 = 0
Now we can factor the equation. We need to find two numbers that multiply to -306 and have a sum of 1:
(x + 18)(x - 17) = 0
From here, we have two options:
Option 1: x + 18 = 0
If x + 18 = 0, then x = -18, which is not a positive integer. Therefore, we discard this option.
Option 2: x - 17 = 0
If x - 17 = 0, then x = 17.
Therefore, the two consecutive positive integers are 17 and 18.
To find their sum, we add the two integers together:
Sum = 17 + 18 = 35
So, the sum of the two consecutive positive integers is 35.
To solve this problem, we need to set up an equation based on the given information. Let's suppose the first positive integer is x.
According to the property given, the second positive integer is x + 1 (consecutive integers differ by 1).
Now, we can set up the equation based on the property: x * (2 * (x+1)) = 612.
Expanding the equation: 2x * (x + 1) = 612.
Simplifying further: 2x^2 + 2x = 612.
Rearranging the equation: 2x^2 + 2x - 612 = 0.
To solve this quadratic equation, we can either factorize or use the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a.
For our equation, a = 2, b = 2, and c = -612.
Substituting these values into the quadratic formula, we get:
x = (-(2) ± √((2)^2 - 4(2)(-612))) / (2(2)).
Simplifying further:
x = (-2 ± √(4 + 4896)) / 4.
x = (-2 ± √(4900)) / 4.
x = (-2 ± 70) / 4.
Now, we have two possible solutions:
x₁ = (-2 + 70) / 4 = 68 / 4 = 17.
x₂ = (-2 - 70) / 4 = -72 / 4 = -18.
Since we're looking for positive integers, we discard the negative solution x₂ = -18.
Therefore, the first positive integer is 17, and the second positive integer is 17 + 1 = 18.
The sum of these two integers is 17 + 18 = 35.
So, the sum of the consecutive positive integers is 35.