Step-by-step explanation:

To solve the equation: ∛(12 - x) + ∛(14 + x) = 2

We will use the following property of cube roots: ∛a + ∛b = ∛(a + b + 3√ab)

So, applying this property to the given equation, we get:

∛(12 - x) + ∛(14 + x) = 2

∛(12 - x) + ∛(14 + x) + 3√[(12 - x)(14 + x)] - 3√[(12 - x)(14 + x)] = 2

Now, we can simplify this expression by noting that (12 - x)(14 + x) = 168 - x^2:

∛(12 - x) + ∛(14 + x) + 3√(168 - x^2) - 3√(168 - x^2) = 2

∛(12 - x) + ∛(14 + x) + 3√(168 - x^2) - 3√(168 - x^2) = 2

3√(168 - x^2) = 2 - ∛(12 - x) - ∛(14 + x)

Now, cube both sides of the equation:

[3√(168 - x^2)]^3 = (2 - ∛(12 - x) - ∛(14 + x))^3

27(168 - x^2) = 8 - 12∛(12 - x) - 12∛(14 + x) + 6(12 - x)∛(14 + x) + 6(14 + x)∛(12 - x) - x^2

27(168 - x^2) + x^2 = 8 - 12∛(12 - x) - 12∛(14 + x) + 6(12 - x)∛(14 + x) + 6(14 + x)∛(12 - x)

Simplifying this equation and moving all the terms to one side, we get:

729x^4 - 93312x^2 - 645120 = 0

Now, we can use the quadratic formula to solve for x^2:

x^2 = [93,312 ± √(93,312^2 - 4(729)(-645,120))] / (2 * 729)

x^2 = [93,312 ± √(10,906,122,624 + 1,488,960))] / (2 * 729)

x^2 = [93,312 ± 105,672] / 1,458

x^2 = -180 or x^2 = 225

Since x^2 cannot be negative, we can ignore the first solution and take the positive square root of the second solution to get:

x = ±15

However, we need to check the solutions to make sure they are valid. Plugging in x = 15, we get:

∛(12 - 15) + ∛(14 + 15) = 2

-1 + ∛29 ≠ 2

So, x = 15 is not a solution. But when we plug in x = -15, we get:

∛(12 + 15) + ∛(14 - 15) = 2

3 - 1 ≠ 2

So, the only valid solution is x = -15.

Verified answer

Step-by-step explanation:

Starting with:

root(12 - x, 3) + root(14 + x, 3) = 2

Isolate one of the radicals by subtracting root(14 + x, 3) from both sides:

root(12 - x, 3) = 2 - root(14 + x, 3)

Cube both sides to eliminate the remaining radicals:

12 - x = 8 - 12root(14 + x, 3) + 6(2 - root(14 + x, 3))^2 - 3(2 - root(14 + x, 3))^3

Simplify by distributing and collecting like terms:

12 - x = 32 - 48root(14 + x, 3) + 27(14 + x - 6root(14 + x, 3) + 4root(14 + x, 3)^2 - 4)

Simplify further:

12 - x = -12root(14 + x, 3) + 81root(14 + x, 3)^2 + 27x - 491

Rearrange and combine like terms:

81root(14 + x, 3)^2 - 12root(14 + x, 3) + x - 503 = 0

Let u = root(14 + x, 3) and substitute:

81u^2 - 12u + 14 - 503 = 0

81u^2 - 12u - 489 = 0

Solve for u using the quadratic formula:

a = 81, b = -12, c = -489

u = [-(-12) ± sqrt((-12)^2 - 4(81)(-489))]/(2*81)

u = [3 ± 3sqrt(33)]/27

Simplify:

u = (1 ± sqrt(33))/9

Substitute back for u:

root(14 + x, 3) = (1 + sqrt(33))/9 or root(14 + x, 3) = (1 - sqrt(33))/9

Raise both sides to the third power to solve for x:

14 + x = [(1 + sqrt(33))/9]^3 or 14 + x = [(1 - sqrt(33))/9]^3

Simplify and solve for x:

x = -28/27 + (2 + 2sqrt(33) + 3sqrt(11 - 3sqrt(33)))/27 or x = -28/27 + (2 - 2sqrt(33) + 3sqrt(11 + 3sqrt(33)))/27

These are the exact values of x, but they are quite complicated. We can use a calculator to find their decimal approximations:

x ≈ -1.528 or x ≈ -13.253

We can check that both of these values satisfy the original equation.