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Let a and b be real numbers such that a^3 + 3ab^2 = 679 and 3a^3 - ab^2 = 615. Find a - b.

 Apr 13, 2024
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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)

 Apr 13, 2024

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