If x³ = 184 + y³ and x = 4 + y, then the value of (x + y) is (given that x > 0 and y > 0)

Given:

x3 = 184 + y3

x = 4 + y

x > 0 and y > 0

Calculation:

Substitute x = 4 + y into x3 = 184 + y3

(4 + y)³ = 184 + y³

Expand (4 + y)³ :

(4 + y)3 = 4³ + 3 × 4² × y + 3 × 4 × y² + y³

64 + 48y + 12y² + y³ = 184 + y³

Subtract y³ from both sides:

64 + 48y + 12y² = 184

Rearrange the equation:

12y² + 48y + 64 = 184

Subtract 184 from both sides:

12y2 + 48y + 64 - 184 = 0

12y2 + 48y - 120 = 0

Divide by 12:

y2 + 4y - 10 = 0

Solve the quadratic equation y2 + 4y - 10 = 0 using the quadratic formula:

\(y = \frac{-b \pm √{b^2 - 4ac}}{2a}\)

where a = 1 , b = 4 , and c = -10

\(y = \frac{-4 \pm √{4^2 - 4 × 1 × (-10)}}{2 × 1}\)

\(y = \frac{-4 \pm √{16 + 40}}{2}\)

\(y = \frac{-4 \pm √{56}}{2}\)

\(y = \frac{-4 \pm 2√{14}}{2}\)

\(y = -2 \pm √{14}\)

Since y > 0 , we take the positive root:

y = -2 + √14

Now, find x :

x = 4 + y

x = 4 + (-2 + √14)

x = 2 + √14

Finally, find x + y :

x + y = (2 + √14) + (-2 + √14)

x + y = 2√14

The value of x + y is 2√14.