Find all pairs of integers whose difference is 5 and product is 234.
The square root of 234 is a little over 15. The two integers must be near 15. I played around a little with this -- and sure enough -- it didn't take me long to figure out the two numbers are 13 and 18.
To find all pairs of integers that satisfy the given conditions, we can use a systematic approach.
Let's assume the two integers we need to find are a and b.
1. First, we know that the difference between the two integers is 5. So we can set up the equation:
a - b = 5 (Equation 1)
2. We also know that the product of the two integers is 234. So we can set up the equation:
a * b = 234 (Equation 2)
Now, we have a system of two equations (Equations 1 and 2) with two variables (a and b). We can solve these equations simultaneously to find the values of a and b.
To do that, we can use the substitution method. Let's solve Equation 1 for a:
a = b + 5 (Equation 3)
Now substitute Equation 3 into Equation 2:
(b + 5) * b = 234
Expanding the equation:
b^2 + 5b = 234
Rearranging the equation to the standard quadratic form:
b^2 + 5b - 234 = 0
Now, we can solve this quadratic equation to find the possible values of b. We can either factorize the quadratic or use the quadratic formula. Let's use the quadratic formula:
b = (-5 ± sqrt(5^2 - 4*1*(-234))) / 2*1
Simplifying the equation:
b = (-5 ± sqrt(25 + 936)) / 2
b = (-5 ± sqrt(961)) / 2
b = (-5 ± 31) / 2
This gives us two possible values for b:
b = (-5 + 31) / 2 = 13
b = (-5 - 31) / 2 = -18
Now, substitute these values of b back into Equation 3 to find the corresponding values of a:
For b = 13:
a = 13 + 5 = 18
For b = -18:
a = -18 + 5 = -13
Therefore, the pairs of integers that satisfy the given conditions are (18, 13) and (-13, -18).