+1839 Let a and b be real numbers such that a^3 + 3ab^2 = 679 and 3a^3 - ab^2 = 615. Find a - b.
+1234 We can solve this system of equations by adding and subtracting the equations strategically to eliminate one of the variables.
Adding the Equations:
Adding the two given equations eliminates the term ab^2:
(a^3 + 3ab^2) + (3a^3 - ab^2) = 679 + 615
Combine like terms:
4a^3 = 1294
Solving for a^3:
Divide both sides by 4:
a^3 = 1294 / 4 = 323.5
Subtracting the Equations:
Subtracting the second equation from the first equation eliminates the term 3a^3:
(a^3 + 3ab^2) - (3a^3 - ab^2) = 679 - 615
Combine like terms:
4ab^2 = 64
Solving for ab^2:
Divide both sides by 4:
ab^2 = 16
Relating a and b:
Since we know both a^3 and ab^2, we can try to express one variable in terms of the other.
From the equation for a^3, we can write a =∛(323.5).
Relating a - b:
We want to find a - b. Since we don't have a direct equation for b, we can try to manipulate the equation for ab^2.
Rewrite the equation for ab^2:
b^2 = a(ab^2) = a * 16
Substitute a = ∛(323.5):
b^2 = ∛(323.5) * 16
Now, we can express b in terms of a:
b = ± 4 * ∛(323.5)
Finding a - b:
Since a and b are real numbers, we can consider both positive and negative values of b. However, we only care about the difference a - b.
There are two cases:
Case 1: b = 4 * ∛(323.5)
a - b = ∛(323.5) - (4 * ∛(323.5))
Factor out ∛(323.5):
a - b = ∛(323.5) (1 - 4)
a - b = -3 * ∛(323.5)
Case 2: b = -4 * ∛(323.5)
a - b = ∛(323.5) - (-4 * ∛(323.5))
Factor out ∛(323.5):
a - b = ∛(323.5) (1 + 4)
a - b = 5 * ∛(323.5)
Conclusion:
Since we don't know the signs of a and b beforehand, both cases are valid. Therefore, a - b can be either -3 * ∛(323.5) or 5 * ∛(323.5).
Both answers are negative and positive multiples of the same cube root, so they essentially represent the same value with opposite signs. The absolute value of a - b is:
|a - b| = |(-3) * ∛(323.5)| = |5 * ∛(323.5)| = 5 * ∛(323.5)
