Let \(a\) and \(b\) be real numbers such that \(a^3 + 3ab^2 = 679\) and \(3a^2 b + b^3 = 615\) Find \(a - b\)
We will use the given equations to solve for a and b, and then compute a - b.
From the first equation, we have:
a^3 + 2ab^2 = 679
We can rearrange this equation to isolate a^3:
a^3 = 679 - 2ab^2
Next, we can substitute this expression for a^3 into the second equation, giving:
3a^2b + b^3 = 615
3*(679 - 2ab^2)*b + b^3 = 615
Expanding and simplifying, we get:
2037b - 6ab^3 + b^3 = 615
6ab^3 - b^3 + 2037b = 615
Factoring out b, we get:
b(6a^2 - 1 + 2037) = 615
b(6a^2 + 2036) = 615
b(3a^2 + 1018) = 307.5
Similarly, we can substitute the expression for b^3 from the second equation into the first equation:
a^3 + 2ab^2 = 679
a^3 + 2a*(615 - 3a^2b) = 679
a^3 + 1230a - 1845a^3*b = 679
(1 - 1845b)a^3 + 1230a = 679
a((1 - 1845b)a^2 + 1230) = 679
a^2 = 679/(1 - 1845b) - 1230/(1 - 1845b)^2
Now we can substitute this expression for a^2 into the equation for b(3a^2 + 1018) = 307.5, giving:
b(3*(679/(1 - 1845b) - 1230/(1 - 1845b)^2) + 1018) = 307.5
Simplifying this equation, we get:
b(-2615b^3 + 4071b^2 - 1896b + 517) = 0
This equation has four solutions for b, but only one of them is real, namely:
b ≈ 0.693
We can substitute this value of b back into the equation for a^2, giving:
a^2 ≈ 205.5
Taking the square root of this value, we get:
a ≈ 14.33
Finally, we can compute a - b:
a - b ≈ 14.33 - 0.693 ≈ 13.64
Therefore, a - b ≈ 13.64.
