+5 Let and 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 and a and b
(a) We can use the difference of cubes factorization to solve for ab:
(a−b)(a2+ab+b2)=52
Since a−b=4, we can substitute this value to get:
4(a2+ab+b2)=52
Solving for a2+ab+b2, we get:
a2+ab+b2=13
We can then use the quadratic formula to solve for ab:
ab=2a−b±b2−4ac
Substituting a2+ab+b2=13 and a−b=4 into the quadratic formula, we get:
ab=2⋅4−4±42−4⋅13⋅4
ab=8−4±−112
ab=8−4±4isqrt(7)
Therefore, the possible values of ab are −2isqrt(7),2isqrt(7),−4,4.
(b) We can use the sum of cubes factorization to solve for a+b:
(a+b)(a2−ab+b2)=a3+b3
Since a3−b3=52, we can substitute this value to get:
(a+b)(a2−ab+b2)=52
Solving for a+b, we get:
a+b=a2−ab+b2a3−b3=1352=4
(c) Since we already found that ab can take on the values −2i7sqrt(7),2isqrt(7),−4,4, and a+b=4, we can find all possible values of a and b by substituting these values into the equation a−b=4.
For ab=−2i7, we get:
a−(−2i7)=4
a=4−2i7
For ab=2i7, we get:
a−(2i7)=4
a=4+2i7
For ab=−4, we get:
a−(−4)=4
a=8
For ab=4, we get:
a−4=4
a=8
Therefore, the possible values of a and b are (4−2isqrt(7),8),(4+2isqrt(7),8),(8,8).
