The difference between two positive integers is 3. If the smaller is added to the square of the larger, the sum is 417. Find the integers
smaller --- x
larger --- x+3
(x+3)^2 + x = 417
x^2 + 6x + 9 + x - 417 = 0
x^2 + 7x - 408 = 0
factor, then solve for x, reject the negative value for x
let x = smaller , y = larger
y - x = 3 ... y - 3 = x
y^2 + x = 417
solve the system
Let's call the smaller integer x, and the larger integer y.
According to the given information, we know that the difference between the two positive integers is 3, so we can set up an equation:
y - x = 3 (Equation 1)
We are also told that if the smaller integer is added to the square of the larger integer, the sum is 417:
x + y^2 = 417 (Equation 2)
To solve for x and y, we can use the first equation to express x in terms of y:
x = y - 3
Now substitute this expression for x in Equation 2:
(y - 3) + y^2 = 417
Expand the equation:
y - 3 + y^2 = 417
Rearrange the equation:
y^2 + y - 420 = 0
This is a quadratic equation, so we can solve it by factoring or using the quadratic formula. Since it can be factored, we will factor it:
(y - 20)(y + 21) = 0
By setting each factor equal to zero, we get:
y - 20 = 0 or y + 21 = 0
Solving for y gives:
y = 20 or y = -21
Since y represents the larger integer, we can discard the negative solution. Therefore, the larger integer is 20.
To find the smaller integer, substitute this value of y into Equation 1:
20 - x = 3
Solving for x gives:
x = 20 - 3 = 17
So, the smaller integer is 17 and the larger integer is 20.
To find the integers, let's assign variables to them. Let's call the smaller integer x and the larger integer y.
From the given information, we know that the difference between the two positive integers is 3. So we can set up an equation:
y - x = 3 1
We are also given that if the smaller integer is added to the square of the larger integer, the sum is 417. This can be expressed as an equation:
x + y^2 = 417 2
We now have a system of two equations with two variables. We can solve this system to find the values of x and y.
First, let's solve equation 1 for y:
y = x + 3
Now substitute this expression for y in equation 2:
x + (x + 3)^2 = 417
Expand and simplify:
x + (x^2 + 6x + 9) = 417
x + x^2 + 6x + 9 = 417
x^2 + 7x + 9 = 417
Rearrange to set the equation equal to zero:
x^2 + 7x + 9 - 417 = 0
x^2 + 7x - 408 = 0
Now we can solve this quadratic equation to find the values of x. We can factor it or use the quadratic formula.
Factoring the equation, we find:
(x + 24)(x - 17) = 0
Setting each factor to zero and solving for x, we have:
x + 24 = 0 or x - 17 = 0
Solving each equation, we get:
x = -24 or x = 17
Since we're looking for positive integers, we can discard the negative solution. Therefore, x = 17.
Now substitute this value back into equation 1 to find y:
y = x + 3
y = 17 + 3
y = 20
So the two positive integers are 17 and 20. These are the values of x and y, respectively.