Let \(a\) and \(b\) be real numbers such that \(a-b=4\) and \(a^3 - b^3 = 52\)
(a) Find all possible values of \(ab\)
(b) Find all possible values of \(a+b\)
(c) Find all possible values of \(a\) and \(b\)
To solve the problem of Let a and b be real numbers such that a - b = 4 and a^3 - b^3 = 52 1, we can use the following approach:
(a) We can use the formula for the difference of cubes to factor a^3 - b^3 = (a - b)(a^2 + ab + b^2) and substitute a - b = 4 to get:
52 = 4(a^2 + ab + b^2) 13 = a^2 + ab + b^2
We can then use the formula for the sum and product of roots to find the possible values of ab, given that a + b = (a - b) + 2b = 4 + 2b:
a + b = 4 + 2b a^2 + ab + b^2 = 13
Substituting a = 4 + 2b - b and simplifying, we get:
b^2 - 6b + 9 = 0 (b - 3)^2 = 0
Thus, b = 3 and a = 7, so ab = 21 is the only possible value.
(b) We can use a - b = 4 to express a as a function of b, and substitute into the equation a + b = 2a - 4 to get:
a + b = 2(a - 2) + b = 2b
Therefore, a + b can be any even number.
(c) From part (a), we know that a = b + 4, so substituting into the equation a^3 - b^3 = 52 and simplifying, we get:
3b^2 + 12b + 16 = 0 b^2 + 4b + 16/3 = 0 (b + 2)^2 + 8/3 = 0
This equation has no real solutions, so there are no possible values of a and b other than a = 7 and b = 3 that satisfy the given conditions.
Therefore, the answer is: (a) ab = 21. (b) a + b can be any even number. (c) a = 7 and b = 3 are the only possible values.